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### Rev Politéc. (Quito) vol.42 no.1 Quito Ago./Out. 2018

Articles

Qualitative Behaviour Analysis of Feedback-Controlled Buck-Boost Power Converters Thru Three Different Techniques

Análisis Cualitativo del Comportamiento del Convertidor de Potencia Buck-Boost por Realimentación del Vector de Estado a Través de Tres Diferentes Técnicas

2Departament de Matemàtiques, Universitat Autónoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, España

3Universidad de Los Andes, Ing. de Sistemas, Dpto. de Sistemas de Control, Mérida 5101, Venezuela

Abstract:

This work is based on the comparison of three techniques for analyzing the qualitative behaviour of nonlinear dynamic systems, including the study of their finite and infinite equilibrium points. The qualitative techniques used are: the direct method of Lyapunov, the theorems of Dickson and Perko for second order quadratic differential systems and the linearization around finite equilibrium points. These techniques provide information about the local or global stability of nonlinear systems. The state feedback controlled Buck-Boost power converter will be used as a case of study.

Keywords: Nonlinear system; Bounded system; Qualitative analysis of dynamical systems; Lyapunov method; Buck-Boost power converter

Resumen:

Este trabajo se basa en la comparación de tres técnicas de análisis del comportamiento cualitativo de sistemas dinámicos no lineales, incluyendo el estudio de los puntos de equilibrio finitos e infinitos. Las técnicas cualitativas utilizadas son: el Método Directo de Lyapunov, los Teoremas de Dickson y Perko para sistemas cuadráticos de segundo orden y la Linealización alrededor de los puntos de equilibrio finitos. Estas técnicas aportan información respecto a la estabilidad global o local del sistema no lineal. Como sistema dinámico no lineal se utilizará el convertidor de potencia Buck-Boost realimentado por medio del vector de estados.

Palabras clave: Sistemas No Lineales; Sistemas acotados; Análisis cualitativo de sistemas dinámicos; Técnicas de Lyapunov method; Convertidor de potencia Buck-Boost

1. INTRODUCTION

It is known that some of the inherent qualitative characteristics of dynamic systems have been specified through rigorous analytic techniques. However in the specific case of nonlinear systems, there may be examples where there are not explicit solutions for the differential equations that describe their dynamics, and further, there are systems that exhibit multiple equilibrium points, limit cycles, bifurcations, among other features. Under these circumstances, the qualitative analysis of differential equations is a viable alternative to learn about the dynamic behaviour of these systems.

In this sense, the converse theorems are key tools in the stability analysis of dynamic systems. Some classical references on the subject are the works reported in Krasovskii (1963) and Hahn (1967). More recent references are the papers presented in Khalil (2000) and Fantoni and Lozano (2002). The references mentioned have been developed as a result of the research effort published in Lyapunov (1892); where the local and global equilibrium points in linear systems and in some nonlinear systems are studied. A concise reference to the concepts of the theory of Lyapunov is the text by Slotine and Li (1991).

In general, local results do not provide a comprehensive explanation of the behaviour of nonlinear systems. Therefore it is necessary to use other tools for the study of systems of second order quadratic differential equations, as the one considered in this paper. For this purpose, two references that analyze the behaviour of these differential equations are used: the first is aimed at sorting through the use of inequalities the different behaviours of bounded quadratic systems Coppel (1966), and the second is the work shown in Perko (1970) which, through qualitative analysis of these dynamic systems, seeks to classify them in terms of an atlas represented in phase portraits. Both references are summarized in the textbook Perko (2000).

An application to stability analysis through the qualitative techniques referenced above is presented in Spinetti-Rivera (2011), where the behaviour of the Boost power converter is discussed.

The analysis technique presented in Spinetti-Rivera (2011) was validated by answering the questions presented in the work of Sira-Ramírez (2005), on the stability of the operation in some power converters. This analysis technique was complemented and improved in successive works published by Spinetti-Rivera et al. (2015) and Llibre et al. (2015). Some of the tools used in this work for qualitative analysis of quadratic closed loop feedback systems were obtained from the work Coll et al. (1987) and Giacomini et al. (1996).

This paper develops a qualitative analysis of a nonlinear closed-loop system, specifically the Buck-Boost power converter with a state vector feedback. It is a second order non-linear quadratic differential system that has no explicit solution. It is intended to study the behaviour of the trajectories between finite and infinite equilibrium points, with respect to changes in system and controller parameters, specifically when there is a single finite equilibrium point; for which three techniques are used, namely, the direct method of Lyapunov, the theorems of Dickson and Perko for second order quadratic differential systems and the linearization around finite equilibrium points.

2. BUCK-BOOST POWER CONVERTER

The main characteristic of the Buck-Boost circuit design can be operate as a step up or as a step down voltage converter, that is, its output voltage may be lower or higher than the power supply. Figure 1 illustrates its circuit diagram.

The paper in Sira-Ramírez (1988) shows how the average behaviour of the circuit of Figure 1 may be represented by a continuous time model, applying the laws of Kirchhoff and Ohm. This is:

Ldidt=1-uv+uE, Cdvdt=-1-ui-vR (1)

where i is the inductor current, v the voltage on the capacitor, R the resistance of the load, L the inductance of the coil, C the capacitance, E the power supply and u the DC control input, which is defined in the range [0,1]. In order to facilitate the calculations, let τ and Q be defined as τ = tLC and Q=RCL, and let the linear transformation given in equation 2 be applied to system (1)

xy=1ECL001Eiv (2)

The normalized system (1) may be represented as

dxdτ=x˙=1-uy+u, dydτ=y˙=-1-ux-yQ. (3)

where the normalized variable x(τ)=x is the coil current, y(τ)=y is the capacitor voltage, Q is the charge and u [0,1] is the control input. The equilibrium points of the open loop system (3) are given as

x-=y-(y--1)Q, u-=y-y--1 (4)

wherein the desired value of the output voltage of the capacitor Vd is equilibrium value of the system, that is, y- =VdE<0.

By moving system (3) to the origin, which is achieved through the change of coordinates defined by e1=x-x-, e2=y-y-, eu=u-u-, the following exact error dynamics is obtained

e˙1=1-eu- u-e2+y-1-eu-u-+eu+u-, e˙2=eu+ u--1e1-e2Q-x-1-eu-u--y-Q (5)

Before making the analysis of equilibrium points, the feedback control loop using the state vector with a gain k=[α β] will be considered. Thus, the equation of the control law is defined by

eu=-ke1e2=-(e1+βe2) (6)

The dynamics of the closed loop system is obtained replacing the controller (6) in (5) and the equilibrium values of x- and u- given in (4). That is

e˙1=y--1e1+βy--12-1y--1e2+e1e2+βe22, e˙2=Q-y-y--12y--1Qe1-βy-y--1+1Qe2-βe1e2-e12 (7)

Equating to zero the right sides of (7), solving for e1 in the first equation and substituting it into the second equation, the equilibrium equation based on e2- is obtained

pe-2=e-23+βQ+3y--2e-22+Qβy--12-1+y--12(2y--1)(y--1)e-2=0 (8)

Since (8) is a polynomial of degree three, the Cardano method to characterize its roots will be used. The discriminant of Cardano C is as follows

C=Qβy--12-1+y--122y--12(Q2y--1β2+2Qy--1y-β+2y--1y-2+4Q) (9)

• If C>0, there will be a real root and two complex non-real roots.

• If C=0, there will be a double real root and a single real root.

• If C<0, there will be three real roots.

This work will analyze only the case when there is one real equilibrium point; for which Proposition 1 establishes the range of values for the parameters.

Proposition 1. The existence of a single real equilibrium point is defined by the conditions

a>0 and -y-Q-2-Qy--1<β<-y-Q+2-Qy--1

Restrictions for the parameters are: Q>0, y- < 0, α, β (-∞,∞). The equation of Cardano states that if C then there will be a single real equilibrium point. Thus, from Equation (9) the following set of inequality solutions is obtained

• α>0

• β1<β<β2, where β1,2=-αy-Q2-Qy--1

This range is defined as

RC ={(,β)R| > 0β1<β<β2}

3. LYAPUNOV STABILITY ANALYSIS

In order to apply the concept of stability in the sense of Lyapunov, under the conditions provided in Proposition 1, it is necessary to have a single real equilibrium point located at the origin (0,0).

Theorem 1. System (7) is globally stable if there is a unique equilibrium point.

Proof. Let the positive definite Lyapunov function candidate V(e) be defined as

Ve=12e1e2T1001e1e2 (10)

According to the direct method of Lyapunov, if the derivative of the Lyapunov function candidate evaluated in the trajectories of the dynamical system is negative definite, then the system will display a globally stable behaviour. The derivative V˙(e) is defined as

V˙e=e1e2T1001e1˙e2˙

Substituting Equation (7) in (10) yields an error dependent equation, which has the form V˙e=eTMe+eTKe, where M is a symmetric matrix and K is a skew symmetric matrix defined as follow

M=y--1βy--12-1y--1Q-y-(y--1)2(y--1)Q-βy-y--1+1Q,

K=0e1+βe2-e1-βe20

Since V˙e is the sum of two quadratic forms and eTKe=0, by decomposing the matrix M in a symmetric matrix plus an skew symmetric matrix, i.e. M=Ms+Ma, the derivative of the Lyapunov function candidate turns into V˙e=eTMse+eTMae, where

MS=y--112y--1(Qβ-y-)Q12y--1(Qβ-y-)Qβy-y--1-1Q,

and Ma is the skew symmetric matrix of Ms. Since the term eTMae=0, it suffices analyzing the function V˙e=eT(Ms)e.

Rewriting V˙e, the derivative of the Lyapunov function candidate takes the form V˙e=eT(-Ms)e, namely

V˙e=-e1e2T-y--1-12y--1(Qβ-y-)Q-12y--1(Qβ-y-)Q-βy-y--1-1Q

where the sign of V˙e depends on the sign of the matrix -Ms.

According to Sylvester criterion it suffices that -Ms>0, so that V˙e<0. In order to satisfy that -Ms>0, the minors of the matrix -Ms must have positive determinants. These determinants are defined as:

• The determinant of the first minor is 1=-y--1.

• The determinant of the matrix -Ms is given as 2=β2y--1Q2+2βy-2-βy-+2Q+2y-2y--1

To ensure compliance of 1>0, it is necessary that -y--1>0, which is true if and only if α>0, because by definition y-<0. For 2>0, it is necessary to write it in terms of a quadratic polynomial defined as

pβ=β2Q2y--1+β2Qy-y--1+4Q+2y-2y--1>0, (11)

since the term that accompanies β2 is negative definite (Q2y--1<0), then the polynomial p(β) is positive definite within the interval of its solutions

-y-Q-2-Qy--1<β<-y-Q+2-Qy--1

Note that since of Q>0, α>0 and y-<0, then the determinant of Equation (11) satisfies p(β)=-Qy--1>0; implying that β1,2 R and there will always exist an interval (β12) in which p(β)>0 and therefore, the interval where V˙e<0 is defined by the set RL, given by

RL={(,β)R|>0β1<β<β2} (12)

From the above analysis it is shown that V˙e<0 on RL defined by (12). Also, RL=RC when there is a unique point of equilibrium and therefore this equilibrium point is globally stable.

Corollary 1. In the boundary conditions for the System (7) stability in the sense of Lyapunov does not apply.

Proof. If the discriminant of Cardano C is analyzed using equation (9), it may be appreciated that if β=β1 or β=β2 then C=0, which implies that there are two equilibrium points, and therefore the concept of global stability in the sense of Lyapunov cannot be applied.

4. ANALYSIS VIA THEOREMS OF DICKSON AND PERKO

Theorems of Dickson and Perko (1970), see Appendix, allow qualitative analysis of quadratic second order systems. Theorem 5 is formulated to analyze Bounded Quadratic Systems ($BQS$), while Theorem 6 facilitates studying the qualitative behaviour of systems with a unique real equilibrium point (BQS1).

According to Theorem 5 of Appendix, after applying a linear transformation, System (7) must be affine and equivalent to one of the following Systems (25), (26) or (27).

Theorem 2. The quadratic system described by (7) is bounded.

Proof. Consider System (7) and the linear matrix transformation defined by e = θz

e1e2=θ11θ12θ21θ22e1e2, (13)

where θ11, θ12, θ21 and θ22 are constants. Substituting Equation (13) into (7) the following system of differential equations is obtained

z1˙=Θ1Z12+Θ2Z1Z2+Θ3Z22+Θ4Z1+Θ5Z2, z2˙=Θ6Z12+Θ7Z1Z2+Θ8Z22+Θ9Z1+Θ10Z2, (14)

Where Θ1,...,Θ10 are parameters which depend on θ11, θ12, θ21, θ22, Q, y-, α and β. Equation (14) may be rewritten in the form of the system of equations (27). This is

z1˙=a11z1+a12z2+z22,  z2˙=a21z1+a22z2-z1z2+cz22 (15)

In order to accomplish this transformation, it is necessary to select Θ1 = 0, Θ2 = 0, Θ3 = 1, Θ6 = 0, Θ7 = - 1 and to solve the system of algebraic equations which result in

θ11=ββ2+2, θ12=θ21=-2+β2, θ22=-ββ2+2 (16)

By replacing the coefficients (16) in (14), the parameters of system (15) are

a11=Θ4=-22+β2Q, a12=Θ5=2+β2βy--12-1Q+3y--12y-2+β21-y-Q+βy--11+βy--1y-2+β21-y-Q, a21=Θ9=Q2+β2+βy--12+β21-y-Q, a22=Θ10=-β22+β2Q+y--1-βy--1y-Q , c=Θ8=0 (17)

Since Q>0 then a11<0 and therefore, according to Theorem 5, systems (15) and (7) are bounded.

It should be noted that Theorem 2 ensures that system (7) is bounded for any configuration of finite and infinite equilibrium points. Figure 2 shows the equilibrium points at infinity in a saddle--node configuration, where the circle corresponds to the neighborhood of infinite.

4.2 Bounded quadratic systems with a unique real equilibrium point (BQS1)

Theorem 6 of the appendix allows analyzing the different qualitative behaviours when there is a unique finite equilibrium point in a system of the type (27); and does so by means of phase diagrams, including both the finite equilibrium point as those at infinity. According to Theorem 6 there are four configurations, of which (a) and (b) cannot be used because system (7) is affine to a (27) type system, and these are mutually exclusive. Thus, system (7) may be of type (c) or (d). It will be shown that the only feasible configuration for (7) is (c).

Theorem 3. Given the following parameters restrictions:Q>0, y-<0, >0 and-y-Q-2-Qy--1 <β<-y-Q+2-Qy--1, then system (7) has the configuration type (c) of Figure 6.

Proof. According to Theorem 6(c) of the appendix, System (27) will have a unique equilibrium point if the following conditions are satisfied:

i a11<0, iia12-a21+ca112, iiia11+a220 (18)

Theorem 2 allows to satisfy the condition (18)(i). For the second condition, substituting parameters (17) in (18)(ii) results.

f=β2+2Qy-β+Qy-2+4Q(y--1)<0 (19)

Equation (19) is valid in the interval (β1,β2)- where β1,2 = -y-Q2-Qy--1, i.e.,

-y-Q-2-Qy--1<β<-y-Q+2-Qy--1 (20)

The discriminant of f is f = -Qy--1 and β1,2 R if and only if α > 0.

For the third condition, parameters (17) are substituted into Equation (18)(iii) which results in βy--1Q-1y--1y-. This defines an interval [βa,)- where βa=y--1Q-1y--1y-.

Note that (β1,β2)- βa,-=(β1,β2)-, so the range in which (18)(iii) is fulfilled is (20), with α>0, which shows that (7) is a type (c) system. According to Theorem 5, (7) will be a type (d) system if it satisfies the following conditions:

i a11<0, iia12-a21+ca112<4a11a22-a21a12, iiia11+a22>0 (21)

Analogously, Theorem 2 proves that (21)(i) is valid; and the above analysis, in the interval (β1,β2)-, shows that (21)(ii)= (18)(ii)\$. For the third condition, parameters (17) are substituted into (21)(iii) and β<y--1Q-1y--1y- is obtained, which defines an interval (,βb)-, where βb=y--1Q-1y--1y-. The intervals are such that (β1,β2)- ,βb-= and therefore, since condition (21)(iii) is not satisfied, (7) cannot be a type (d) system.

In summary, the set RL given in (12) defines the range within which system (7) is BQS1.

Corollary 2. Since (7) is a BQS1 type (c) system, it is globally stable.

Proof. Since (7) is BQS1 type (c), then there is a unique real equilibrium point to which all trajectories converge; that is, the system is globally stable.

Corollary 3. Since (7) is BQS1 type (c) system, then it has no limit cycles.

Proof. Since (7) is BQS1 type (c) and it cannot be represented as type (d), then there is no limit cycle or periodic solution.

5. QUALITATIVE ANALYSIS AROUND FINITE EQUILIBRIUM POINTS

System (7) will be analyzed locally with respect to its finite equilibrium points using the linearization method that is described in Dumortier et al. (2006), section 1.5.

The procedure consists of two steps; first the finite equilibrium point is obtained. Next, the Jacobian matrix associated with Equation (8) at this equilibrium point is evaluated. Thus the linearized version of the original nonlinear system is obtained. Equations (8) and (9) define the equilibrium equation and the determinant of Cardano C, respectively. The Jacobian matrix A is given as

A=y--1βy--12-1y--1Q-y--12y-Qy--1-βy-y--1+1Q

On the other hand, the eigenvalues of A are defined as

λ1,2=-12Q+y--12-βy--12Qy-4Qy--122y--1+Qy--12β-1+y--12(1+(y--1)(y-β-Q))22Q(y--1) (22)

The local behaviour at the origin of coordinates may be interpreted using the following theorem.

Theorem 4. The origin of coordinates is an attractor.

Proof. The origin of the linearized system is an attractor, if the system possesses a unique equilibrium point and its eigenvalues are negative. This is accomplished with the following restrictions:

i C>0,

iiλ1λ2=Q-y--122y--1-Qy--12βQy--12>0,

iiiλ1+λ2=y--1Q-y-β-1Q<0

The above restrictions are satisfied for the following conditions on the parameter

>0, -y-Q-2-Qy--1<β<-y-Q+2-Qy--1

These conditions are the same that define the set RL given in Equation (12).

In order to verify if the origin is a repeller, restriction (i) in addition to the following restrictions are used

ivλ1λ2=Q-y--122y--1-Qy--12βQy--12>0,

vλ1+λ2=y--1Q-y-β-1Q>0

It is easy to verify that there are not values of parameters that would make the origin to behave as a repeller.

To check if the origin of coordinates is a saddle, the restriction (i) and the following restriction are taken into account

viλ1λ2=Q-y--122y--1-Qy--12βQy--12<0,

There are not values of parameters that would make the origin to behave as a saddle.

To verify if the origin of coordinates is a center, the restriction (i) and the following restrictions are considered

viiReλ1,λ2=0

viiiImλ1,λ20

From (vii) and (22) results Re{λ12}=0 -12Q+y--12-βy--12Qy-=0. Also, from (viii) and (22) results im{λ12}0 4Qy--122y--1+Qy--12β-1+y--12(1+(y--1)(y-β-Q))2<0.

There are not values of parameters that would make the origin to behave as a center.

Theorem 6 may be used to prove that the origin of coordinates, which is the unique real equilibrium point, can only be an attractor; and it is in the RL set defined in Equation (12). Also, the system is locally stable.

6. EXAMPLE 1

For simplicity the standard model (5) is used. For results in the original coordinates it is sufficient to apply the transformation matrix (2). Using equation (6) and errors eu=ut-u-, e1=xt-x-, e2=yt-y- the following controller is obtained

ut=u--axt-x--b(yt-y-) (23)

where the equilibrium points are evaluated using (4). Substituting (23) in the standard model (5), the closed-loop normalized converter dynamics is obtained

x˙t=1+a(xt-x-+byt-y--u-)y(t)-axt-x--b(yt-y-)+u-, y˙t=-1+a(xt-x-+byt-y--u-)x(t)-y(t)Q (24)

Here x(t) and y(t) correspond to the normalized variables inductor current and capacitor voltage, respectively. By selecting the values of the parameters Q=1, a=1, b=1, y-=-2, the coordinates of the equilibrium point turn out to be x-=6 y u-=2/3. The computer simulation is performed with the PPlane-Matlab® program.

In Figure 3a the phase diagram for the variables x,y is shown. The red dot indicates the coordinates (x-,y-)=(6,-2) of the equilibrium point. The corresponding eigenvalues of the equilibrium points are λ1=-0,429 and λ2=-9,5704; ensuring that it is an attractor. The nullclines in Figure 3a are shown in yellow and pink colors. This figure also included three trajectories in blue (ie the trajectory that passes through the origin (0,0)). Figure 3b shows the trajectories x(t) and y(t) with respect to normalized time. It is observed that both the normalized inductor current and the normalized capacitor voltage converge to the coordinates of the equilibrium point (x-,y-)=(6,-2).

7. EXAMPLE 2

In order to visualize realistic effects on the Buck-Boost converter dynamics, the following example considers firstly the use of the average model equation (1), without including the switch and assuming that the input is defined in the continuous range u=[0, 1]. The simulation is performed in Mat Lab, with parameters C=50µF, L=500µH, R=10 ohms and E=10 volts. Figures 5a, 5b and 5c show the current in the inductor, the voltage in the capacitor and the control input. It is shown that with initial conditions i(0)=0 and v(0)=0 the trajectory converges to a single equilibrium (i,v)=(2,-10).

Note: in the average model Equation (1) the equilibrium point is placed in different coordinates of the origin (0,0).

Now, the Buck-Boost converter is simulated using the Orcad-Pspice program to implement the circuit diagram shown in Figure 4. Mathematical models accurately approximate the behaviour of each of the elements that constitute the converter and in this case, a transistor and a diode operate as the discrete switch of the converter, so it is called a switch model. The characteristics that include the circuit diagram implemented with Orcad-Psice are the following: internal resistance of the inductor RL=0.1 ohm, transistor model IRFZ34, diode MUR150, pulse width modulator (PWM) at a frequency of 20 KHz; and the parameters were selected as C=50µF, L=500µH, R=10 ohms and E=10 volts. The control is ideally treated since it can be implemented by means of a microprocessor. Figures 5a, 5b and 5c show the behaviour of each of the variables of the switch model when the initial conditions are i(0)=0 and v(0)=0. The coordinates of the equilibrium point of the switch model correspond to (i, v) = (2,75, -8,5).

In both cases the non-normalized control took the form

u=1ELCi-y-y--1Q-βvE-y-+y-y--1

The differences between the two models are due to the energy losses produced by semiconductors and conductors; i.e. the conductive voltage drops of the semiconductors and the internal resistance of the conductors and semiconductors. The results show the existence of a single equilibrium point.

8. ANALYSIS OF RESULTS

Table 1.  Qualitative analysis results of System (7) with different techniques.

Qualitative Analysis Technique Bounded for any value of the parameters (BQS) Bounded with a unique real equilibrium point (BQS1)
Lyapunov Not shown if bounded Globally Stable
Dickson–Perko Bounded Globally Stable
Linearization Not shown if bounded Locally Stable
Dickson–Perko and Linearization Bounded Globally Stable

There have been used three techniques to study the qualitative behaviour of a second order nonlinear dynamic system. These techniques have corresponded to the direct method of Lyapunov, theorems of Dickson and Perko and the approximate linearization of nonlinear systems. In all three cases, the analysis has led to the same set where the parameters of the system were defined.

The results of the analysis performed with each of the techniques have been summarized in Table Table 1.

• The direct method of Lyapunov allows to demonstrate global stability of System (7) but this does not allow demonstrating boundedness (BQS) for any variation of the parameters.

• \item Theorems 5 and 6 of Dickson and Perko may be used to prove that System (7) is bounded (BQS), regardless of the values of its parameters.

• It also demonstrates that when it exists there is a unique real equilibrium point (BQS1) the System (7) is globally stable and there are no limit cycles in its trajectories.

• Approximate Linearization allows local analysis and provides no information on the overall behaviour of System (7) or its boundedness (BQS). It also demonstrate that when there is a unique real equilibrium point is an attractor and therefore the System (7) is locally stable.

• If the Theorems of Dickson and Perko is using to demonstrate the boundedness feature (BQS) and unique equilibrium point existence. If the approximate linearization method is used then it can be concluded that System (7) is BQS1.

9. CONCLUSIONS

In this work an analysis of the behaviour of the trajectories around the equilibrium points of the Buck--Boost power converter with state vector feedback, using qualitative techniques for dynamic systems has been presented.

The closed loop system has a bifurcation of the equilibrium points. There may exist one, two or three points of finite equilibrium points. In the whole range of the parameters the system is bounded (BQS) for any configuration of finite equilibrium points and there are no limit cycles.

The direct method of Lyapunov can be used to ensure the system global stability. With the Theorems of Dickson and Perko a global qualitative behaviour of the system is obtained with a unique equilibrium point. Both cases provide conditions in the control parameters demonstrate that all trajectories converge to the unique equilibrium point.

Linearization around the origin of coordinates, where it is located the equilibrium point, facilitates establishing conditions on the control parameters to ensure that it is a local attractor.

The direct method of Lyapunov and the Theorems of Dickson and Perko allowed to obtain results on the general behaviour of the system, while the approximate Linearization only allowed to give local results where it is considered as global for the case study using the results of boundedness, together with the absence of limit cycles for the BQS1.

10. THEOREM OF DICKSON AND PERKO

Theorem 5. Any (BQS) is affinely equivalent to

x˙=a11x, y˙=a21x+a22y+xy (25)

with a11 < 0 and a22 0, or

x˙=a11x+a12y+y2, y˙=a22y (26)

with a11 0, a22 0 and a11+a22 < 0, or

x˙=a11x+a12y+y2, y˙=a21x+a22y-xy+cy2 (27)

with |c| < 2 and either (i) a11 < 0; (ii) a11 = 0 y a21 = 0; or (iii) a11 = 0, a21 6= 0, a12+a21 ≠ 0 and ca21+a22 0.

Theorem 6. The phase portrait of any (BQS1) is determined by one of the separatrix configurations in Figure 6. Furthermore, the phase portrait of a quadratic system is given by Figure 6.

(a) iff the quadratic system is affinely equivalent to (25) with a11<0 and a22<0;

(b) iff the quadratic system is affinely equivalent to (26) with a11 < 2a22 < 0;

(c) iff the quadratic system is affinely equivalent to (26) with 2a22 a11 < 0 or (27) with |c| < 2 and either

(i) a11 = a22+a21 = 0, a21 ≠ 0 and a22 < min(0,-ca21) or a22 = 0 < -ca21,

(ii) a11 < 0, (a12-a21+ca11)2 < 4(a11a22-a21a12), and a11+a22 0, or

(iii) a11 < 0 y (a12-a21+ca11) = (a11a22-a21a12) = 0;

(d) iff the quadratic system is affinely equivalent to (27) with |c| < 2 and either

(i) a11 = a12+a21 = 0 and 0 < a22 < -ca21, or

(ii) a11 < 0,a11 + a22 > 0, and (a12 - a21 + ca11)2 < 4(a11a22-a21a12).

Acknowledgements

The second author is partially supported by a MINECO/FEDER grant MTM2008-03437 and MTM2013-40998-P, an AGAUR grant number 2014SGR568, an ICREA Academia, the grants FP7-PEOPLE-2012-IRSES 318999 and 316338, FEDER-UNAB-10-4E-378.

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Published: 31/10/2018

Received: August 07, 2018; Accepted: October 11, 2018

Keiver Sosa. System Engineer and Instructor from the University of Los Andes in Mérida, Venezuela. He is doctorate student in Ciencias Aplicadas and he investigates in the area of control systems, nonlinear power converters and practical applications using digital signal processors.

Jaume Llibre. Full professor at the Autonomous University of Barcelona (Spain), he is a member of the Royal Academy of Sciences and Arts of Barcelona. He was a long term visitor at different important universities and research institutes of several countries. He is the author of many papers and had a large number of Ph.D. students. His main results deal with periodic orbits, topological entropy, polynomial vector fields, Hamiltonian systems and celestial mechanics.

Mario Spinetti-Riverais. Electrical Engineer, Ph.D. in Automation Advanced and Robotic from the University Politécnica de Cataluña at Barcelona, Spain. He is full professor at the University of Los Andes at Mérida (Venezuela). Author of papers in the area of control systems, nonlinear power converters and practical applications using digital signal processors. His research interest are power electronics modeling, nonlinear systems, large-signal analysis and dynamical systems.

Eliezer Colina-Morles. Graduated as a Systems Engineer at the University of Los Andes at Mérida, Venezuela, the degree of Master of Science in Systems Engineering at CaseWestern Reserve University, Cleveland, USA and the degree of Doctor of Philosophy at the University of Sheffield, England. He is Emeritus Professor at the University of Los Andes and presently works in research projects, at the University of Cuenca, Ecuador.