Selection and Validation of Mathematical Models of Power Converters using Rapid Modeling and Control Prototyping Methods

Fredy E. Hoyos1, John E. Candelo2, and John A. Taborda3 Scientific and Industrial Instrumentation Research Group School of Physics, Universidad Nacional de Colombia, Sede Medellı́n, Email: fehoyosve@unal.edu.co Department of Electrical Energy and Automation, Facultad de Minas, Universidad Nacional de Colombia, Sede Medellı́n. Email: jecandelob@unal.edu.co Faculty of Engineering, Electronics Engineering, Universidad del Magdalena, Santa Marta, Magdalena, Colombia. Email: jtaborda@unimagdalena.edu.co


MATHEMATICAL MODEL
A complete schematic diagram of the system under study is shown in Figure 1. The converter is formed by a power source E, an internal source resistor r s , a MOSFET (metal oxide semiconductor field-effect transistor) as a switch S with internal resistance r M , a diode D with direct polarization voltage V f d , a filter LC, an internal resistance of the inductor r L , a resistance used to measure the current r M ed , and a resistance representing the load of the circuit R [16]. The variables measured in the converter are the capacitor voltage υ c and the inductor current i L . These variables are measured in real time and they are sent to the ZAD and FPIC in order to calculate a signal for the centered pulse width modulation (CPWM), which closes the control loop.
This system changes the structure with the action of the switch S, which is managed by the CPWM. This modulator consists of a circuit composed of a switch and a DC power source, which in conjunction with the filter LC and the diode D, must supply an average voltage υ c to the output during a switching period. For this effect, the CPWM changes the switch S between the states ON (E) and OFF (−V f d ). Figure 2 shows the general idea of the CPWM, where d is the duty cycle calculated for each period.
When the control input is u = 1, then the state of the switch S is active (ON) and the system gets into a continuous conduction mode (CCM), which can be modeled as in (1).
This last equation can be simplified and written as in (2).
When the control input is u = 0, switch S is inactive (OFF) and the system can be modeled as shown in (3).
This equation can be simplified and written as in (4).
where a = −1/RC, h = 1/C, m = −1/L, p 2 = −(r s + r M + r M ed + r L )/L, p 3 = −(r M ed + r L )/L, and x 1 = υ c , x 2 = i L . The notation x 1 = υ c represents the capacitor voltage or the voltage at the load bus, and x 2 = i L represents the current through the inductor. The state equations (2) and (4) have been simplified as shown in (5); whereẋ = [ẋ 1 ,ẋ 2 ] = [ dx1 dt , dx2 dt ] . In the input matrices B 1 and B 2 is the information of the control inputs according to the scheme of the CPWM (Figure 2).ẋ =    A 1 x + B 1 if kT ≤ t ≤ kT + dT /2 A 2 x + B 2 if kT + dT /2 < t < kT + T − dT /2 A 1 x + B 1 if kT + T − dT /2 < t < kT + T The next step is to design a control strategy that allows the capacitor voltage (x 1 = υ c ) to be equal to the reference voltage x 1ref or a desire value. To obtain tracking or regulation, the time must be calculated ISSN: 2088-8708 with a predefined period T , in which the switch S must remain closed (u = 1), called the "duty cycle" d, with (d ∈ [0, T ]). Thus, the duty cycle d is defined as the time that the switch S is closed for the period T .

ZAD CONTROL STRATEGY
This control technique was proposed by [17], and tested numerically and experimentally in [15,18,19]. This technique basically consists of defining a function and force an average value of zero at each sampling period. For this particular case, s(t) is used as a function of the state value at the start of the sampling period s(x(kT )). In this case, the function is defined as a linear function ( Figure 3) and slopes are obtained from the values of the state variables in the instant of sampling t = kT as shown in (6) and (7). The function s(x(kT )) is linear in its sections, as shown in Figure 3 and it can be expressed as in (6). whereṡ where k s = K s √ LC and the term K s is a constant of the controller and considered as a parameter in the bifurcation analysis. The condition of the average zero is expressed in (8).
From (8), it is noted that the first and third slopes have the same values. All the information to build s(x(kT )) is obtained from the state values x 1 and x 2 in the instant kT .
Solving the equation related with the condition of average zero (8), the expression for the duty cycle can be expressed as shown in (9).
As in the experimental test, the variables are measured, the data are processed, and a CPWM is calculated with a frequency of 10 kHz with a one-delay period; thus, the expression of the duty cycle is defined IJECE ISSN: 2088-8708 1555 as in (10). This implies that the control law in the current period is calculated with the values of the states measured in the previous iteration.
4. FPIC TECHNIQUE FPIC was first presented in [20]. Later, a numerical test was performed in [19,21] and finally the first experimental results were presented in [12]. In this section, the basis of the FPIC is presented.

FPIC theorem
Consider a system with a set of equations as shown in 11, where: x(t) ∈ R n and f : R n → R n .
Suppose that a fixed point x * exists that is unstable and within the orbit of control; that means x * = f (x * ). Suppose also that J = ∂f ∂x is the Jacobian of the system and that under this condition the system eigenvalues λ i can be calculated. Then, when system is unstable, there is at least one i where |λ i (J)| > 1. Thus, (12) guarantees stabilization in a fixed point when the parameter N has a real positive value.

Demonstration
Initially, it should be noted that in (11), the fixed point has not been altered. In this case, the Jacobian of the new system can be expressed as shown in (13).
where Jc is the Jacobian of the controlled system and J is the Jacobian of the unstable system. Therefore, a correct assignation of the parameter N guarantees stabilization at an equilibrium point, because the eigenvalues of the controlled system will be the eigenvalues of the original system divided by the factor N + 1. One way to calculate directly N is through the Jury stability criterion. Then, by considering the strategy of ZAD and FPIC, a new duty cycle can be calculated with (14).
where d k (kT ) is calculated from (10) and d * . The value is calculated at the start of each period as in (15).
Thus, (14) includes the ZAD and FPIC techniques. Considering that the duty cycle (d) must be greater than zero and less than 1, a new equation that corresponds to the saturation of the duty cycle is shown in (16).

FIRST MODEL OF THE BUCK CONVERTER
The classical model of the buck converter is represented in Figure 4. It consists of a switch, a diode, a filter LC, and a resistance representing the load (R). The DC source used in this case is regulated (E). However, authors of [19] demonstrated that when using the FPIC in the buck converter, the effects of the ISSN: 2088-8708 To obtain the mathematical model represented by equations in the state space, the resulting topologies generated due to the switching must be considered. On the one hand, when u = u 1 = 1, u = u 2 = 0, and the inductor current is positive, then a CCM is presented. On the other hand, when u = u 2 = 0 and the inductor current is zero, then a discontinuous conduction mode (DCM) is presented.
The converter has two energy storage elements (capacitor and inductor) and the state space model has two state variables: the capacitor voltage (υ c ) and the inductor current (i L ). For the case of CCM, the representation of the state space is obtained with (17).
The system described in (17) can be simplified as shown in (18).
where x 1 = υ c , x 2 = i L , a = −1/RC, h = 1/C and m = −1/L. The DCM is presented when the switch is open and the inductor current is equal to zero. In this case, the diode stops conducing and the capacitor is discharged through resistor R. The equation that models the dynamics of this topology is given by (19). It is important to note that although i L = 0, the complete control of the output is not achieved; therefore, the control action is lost until a cycle begins.
Considering that the system operates in CCM, it can be represented asẋ = Ax Because the control signal u has two values u 1 and u 2 , two different topologies for each sampling period are presented. This system is controlled by the CPWM and the model can be expressed as in (20)

Analytical solution for the first model
Some sections of the system shown in (20) are linear and time invariant [17,19]. Thus, each section has a linear system in the formẋ = Ax + Bu, which is solved analytically by using (21).
IJECE ISSN: 2088-8708 1557 After solving each section of (21), the solution in the continuous time is defined as in (22). where: The solution for the system in DCM is given by (24) and is possible when the inductor current is equal to zero.
Starting from the solution in continuous time given in (22) and performing discretization in the output signals for each sampling period T , the following expression in discrete time [19] is given by (25), which is the solution in CCM for the studied converter.
The solution for the system in DCM during the discrete time is given by (26).

ZAD control
Following the procedure described in Section 3. and considering a time delay, the duty cycle with the ZAD is calculated as shown in (27). where:

FPIC control
In the steady state x 1 = x 1ref andẋ 1 =ẋ 1ref = 0. With the last definition, the following consideration is obtained: s(x(t)) = 0. From the first equation of the system,ẋ 1 = ax 1 + hx 2 , is obtained that x 2 = (ẋ 1ref − ax 1ref )/h. Therefore, when regulation is considered, the expressions x * 1 = x 1ref and x * 2 = (ẋ 1ref − ax 1ref )/h for the steady state are calculated. Then, x * 1 and x * 2 are the new state variables, depending only on the reference signal x 1ref and its derivateẋ 1ref .
Replacing x * 1 and x * 2 in (27) and the parameters of the model (17), the duty cycle is calculated as in (29).

ZAD-FPIC control
To control the converter with the ZAD and FPIC techniques, equation (30) is used, where N is the control parameter of the FPIC technique.
Thus, (30) includes both the ZAD (27, 28) and FPIC techniques (29). Considering that the duty cycle must be greater than zero and less than 1, then d can be expressed as in (31).

Stability analysis
This section determines the stability of the periodic orbit 1T for the first model of the buck converter controlled by the ZAD and FPIC with LEs. The LEs are a very powerful tool that helps to determine the convergence of two orbits of a recurrent equation whose initial conditions differ infinitesimally from one another.
Because knowledge of the orbits is required, the analytical calculation becomes very complex. Thus, a numerical procedure is preferred to find them. On one hand, when trajectories are very close to convergence, the associated LEs will be negative. On the other hand, when trajectories diverge, then at least one of the LEs is positive [22]. LEs are directly calculated from the Poincaré application given in (25) and rewritten in (32).
In the functioning scheme with a time delay (n = 1), the system presents four state variables (two current time variables and two delay time variables). This is because the duty cycle d k (kT ) calculated with the ZAD is obtained with the samples measured in (k − 1)T , as shown in (27); thus, when applying the ZAD and FPIC techniques, the following expressions are used for (28)-(31).
Therefore, the solution of the system x(k + 1) = F(x(k)) can be expressed as shown in (33).
where f 1 is the discrete solution in the time for υ c , f 2 is the discrete solution in the time for i L , x 3 (k + 1), and x 4 (k + 1) are the variables υ c and i L in the previous instant (k). The Jacobian of the system is given by (34).
The term q i (DF (x)) is the i-eigenvalue of DF(x(k)). The LE λ i of the respective eigenvalue is given by (35).

SECOND MODEL OF THE BUCK CONVERTER
In this second model, the losses are considered by adding an inductor r L and a resistance used to measure the current r M ed as shown in Figure 5. Therefore, r M ed was considered with an approximate value of 1.007 Ω.
The mathematical model of the CCM is described in (36).
The system (36) can be simplified and expressed as shown in (37), where The system in CCM can be represented similar to the model in (5.), according to the simplified forṁ x = Ax + Bu. As with the simplified model, this system can be represented with the simple equation shown in (38).ẋ In the DCM, the system is modeled in the same way as for the first model of the buck converter as shown in (19). The analytical solutions for the continuous case and the discrete case are the same as the solution for the first model and are given by (22), (24), (25) and (26). In this case, the matrix of the state transition e AT is changed, which is affected by the internal resistances r L and r M ed .

ZAD-FPIC control for the second model of buck converter
The procedure to apply the ZAD technique is the same as described in Section 3. With this procedure, the mathematical expression shown in (39) is obtained. where:

Stability analysis
As in the first model of the buck converter, stability analysis for the second model of the buck converter is performed by using LEs. The procedure is the same as described in Section 5.5.

THIRD MODEL OF THE BUCK CONVERTER
For this model, other types of losses are included for the buck converter model by considering the resistance of the source (r s ) and the resistance of the MOSFET (r M ) as shown in Figure 6. The internal resistance of the source is increased due to the resistances of the contacts, cables, series switch, and shut-down converter.
The MOSFET resistance r M and the forward voltage in the fast diode V f d were taken from datasheets. Additionally, the internal resistance was measured in a laboratory test.
For the control input u = u 1 = 1, the equation in the state space is given as in (42). In a simplified way, this equation can be expressed as in (43). When the switch is open (u = u 2 = 0), the system is modeled as in (44) and simplified as in (45). where For the CCM, these equations have been simplified as shown in (46), where the termẋ = [ẋ 1 ,ẋ 2 ] = [ dx1 dt , dx2 dt ] , B 1 , and B 2 consider the information of the control input such as the voltage source (E) and direct polarization voltage of the diode (−V f d ). Likewise the previous cases, the system controlled by the CPWM operates as expressed in (46).

Analytical solution for the third model of the buck converter
The system operating in CCM has the solution in continuous time given by (47). where: The solution of the system in DCM is given in (49) and it occurs when the inductor current is equal to zero.
Starting from the solution in the continuous time given in (47) and performing discretization in the output signals for each sampling period T , the following expression is obtained in (50), which is a stroboscope map of the solution in CCM for the third model.
The matrix Q is given in (51) and the other expressions are given in (48).
The solution of the system for the DCM in the discrete time is given by (52).

ZAD-FPIC control for the third model of the buck converter
The necessary steps to apply the ZAD control technique are the same described in Section 3. It is important to consider that when the topology changes through the action of the switch, the state matrices of the system and the input arrays also change.
The duty cycle with the ZAD technique is calculated as in (53). where, 1562 ISSN: 2088-8708 When the FPIC control technique is used, then the procedure described in Section 4. must be considered and the equation shown in (55) is obtained.

Stability analysis
As in the first and second model of the buck converter, the stability analysis for the third model of the buck converter is performed by using LEs. The procedure is the same as described in Section 5.5. For this case, the LEs are directly calculated by using the application of Poincaré.

RESULTS AND ANALYSIS
In this section, comparisons between the theoretical, simulation, and experimental tests for the ZAD and FPIC techniques are performed. The theoretical test considers the evaluation of stability with LEs, the simulation test is performed by using the Poincaré map, and the experimental test is performed by using an electronic circuit.

Parameters for the test
The software used for programming the controllers is developed in the control board and DS1104 of dSPACE GmbH. In this board, the ZAD and FPIC were implemented and programmed with the MATLAB-Simulink and ControlDesk software. The controller was implemented in Simulink and uploaded to the DS1104 in order to work as a real-time application. The parameters for all the models are shown in Table 1. All the parameters included were used to build the circuit and compare the results with the models of the theoretical and simulation tests. The effects of the quantization come from the control board and the DS1104, where the controller was implemented with 12 bits for the analogue signals measured in the test υ c and i L . The duty cycle d was considered as 10 bits.   Figure 7b shows the bifurcation diagram obtained with the numerical analysis of the controlled variables υ c versus K s using a Poincaré map. Figure 7c shows the bifurcation diagram for the experimental test with the implemented electronic circuit. The stability behaviors shown for the two first figures were created by changing the control parameter K s from 0 to 100 and maintaining the control parameter N equal to 1. The stability behavior shown in the third figure was created by changing the control parameter K s from 0 to 5 and maintaining the control parameter N equal to 1. The term υ c is the capacitor voltage or the output voltage, the visualization range of which was maintained between 28 V and 36 V. Using this mathematical method, the bifurcation point, the trace of the stability behavior, and the stability zones were detected and plotted (Figure 7a). The point K s = 47.563 represents a stability limit, where the greater values of K s represent the stable zone and the lower values of K s represent the unstable zone.
The simulations confirm that there is a stability zone (right side of Figure 7b) and an instability zone with a limit point close to that determined with the mathematical model. In addition, an unstable zone and chaos are presented when the parameter K s is decreased. The point determined with the theoretical analysis is validated as a point representing the stability limit of the system when K s 50.
Similar to the previous results presented in this paper, the experimental test shows a stable zone (right side of Figure 7c) where the signals remain with low variations, and an unstable zone (left side of the Figure  7c) where the signals change abruptly. Comparison with the two previous figures show that the system can represent similar stability behavior. Besides, the controlled variable υ c is very close to the reference signal υ ref . From this figure, the diagram obtained with the simulation is shifted to the right with respect to the diagram of the experimental test. In addition, the chaos and periodicity zones are expanded on the axis K s as shown in Figure 7b. Figure 7c shows for the experimental test that a stability limit is obtained when the controlled parameter is K s 3.75. Therefore, the three figures show some coincidence and correlation when comparing the responses of the system as they represent similar events. However, the experimental test shows deviations with respect to theoretical and simulated tests when the parameter K s changes. These deviations could be corrected by adding new elements that represent the losses of other elements in the system that are not considered in the simulation model. Figure 8 shows the stability analysis and the comparison between the theoretical, simulation, and experimental tests for the second model of the buck converter controlled by the ZAD and FPIC. As previously described, an internal resistance r L and a resistance of measurement r M ed were added to the model. Figure  8a shows the evolution of the LEs with the mathematical solution. Figure 8b shows the bifurcation diagram obtained with the numerical analysis of the controlled variables υ c versus K s by using a Poincaré map. Figure  8c shows the bifurcation diagram for the experimental test with the implemented electronic circuit. The stability behaviors shown in the three figures were created by changing the control parameter K s from 0 to 5 and maintaining the control parameter N equal to 1. The term υ c is the capacitor voltage or the output voltage, the visualization range of which range was maintained between 28 V and 36 V.

Results for the second model
The  Figure 8. Stability of the second model controlled by the ZAD and FPIC exponents become negative for values of K s > 4.588. Therefore, the analysis determined that for this second model, the stability of the system is found for values of K s < 4.588, which must be validated with other analyses. The bifurcation diagram is shifted to the left of the axis of the parameter K s . The stability limit is represented close to the value obtained with the LEs. However, the signals in the instability zone are larger than the obtained with the mathematical calculation. Therefore, by using the second model presented in this research, the three figures present more coincidence and correlation as they represent better the events when the parameter K s is changed. Compared with the results of the first model, the bifurcations diagrams of the simulation and experimental test are more similar and represent more details of the events. However, the equilibrium point is shifted to the right with respect to the diagram of the experimental tests, although it is improved with respect to the first model. Therefore, by adding the internal resistance of the inductor r L and a resistance for measuring the current r M ed , better similitude is presented between the simulation and experimental tests. Figure 9 shows the stability analysis and the comparison of the theoretical, simulation, and experimental test for the third model of the buck converter controlled by the ZAD and FPIC. As previously described, the additional parameters added to the model were r s , r M , and V f d . First, Figure 9a shows the evolution of the LEs with the mathematical solution. Second, Figure 9b shows the bifurcation diagram obtained with the numerical analysis of the controlled variables υ c versus K s , by using a Poincaré map. Finally, Figure 9c shows the bifurcation diagram for the experimental test with the implemented electronic circuit. The stability behaviors presented in the three figures were created by changing the control parameter K s from 0 to 5 and maintaining the control parameter N equal to 1. The term υ c is the capacitor voltage or the output voltage, the visualization range of which was maintained between 28 V and 36 V. Because most of the losses are included with the inclusion of the parameters r s , r M , r M ed , and r L , a regulated voltage is obtained with an error in the steady state lower than 0.2 %. The bifurcation diagram obtained in the simulation test shows that the stability limit is obtained with a reduced K s lower than that obtained in the second model, which is K s 3.6. Furthermore, it can be concluded that the results obtained IJECE ISSN: 2088-8708 1565

Results for the third model
in the simulation test are more similar to those obtained in the experimental test, but some low differences are still found that will be investigated in future research. The experimental test shows a small cloud of electromagnetic noise produced by the switch commutation; however, this noise can be neglected. In the results obtained by the simulation test, the stability limit is shifted to the right more than in the experimental test. This displacement can be attributed to some parameters not included in the model of the controller such as internal resistance, parasitic capacitances and parasitic inductances presented in the circuit, and the gains of the circuit used for setting the signal.
The results show that the LEs are negative for K s ≥ 3.6 in the theoretical test, which indicates the stability of the system and is similar to the results obtained in the simulation and experimental tests. The stability limit in the experimental test is obtained when K s ≥ 2.7, whereas for the simulation test it is K s ≥ 3.6. The number of bands and their behaviors illustrated in the figures are similar for both the simulation and experimental tests.
In the experimental test, when K s < 2.7, the system slowly loses regulation capacity, starting with chaotic behavior. Next, some periodic bands appear and a new chaos proceeds and, finally, a chaotic band behavior occurs. For both the simulation and experimental tests, the error in regulation increases when the value of K s is decreased.
Because most of resistive losses have been included, the diagrams obtained in simulation are very similar to the experimental test. In this case, it is clear that the controlled variable υ c reaches the reference when K s > 4. Therefore, we can conclude that the third model is the best to represent similar events in the theoretical, simulation, and experimental tests.  Because the bifurcation diagrams have been created with N =1 and K s changing between 0 and 5, the system operates in the stability limits between the fixed points and chaos, as observed in Figures 10a, 10b, and 11a. Although the results show good similitude between the simulation and experimental tests, there are some small differences in magnitude, time, and signal deviation that need to be investigated in future research by considering other parameters.

Two-parameter bifurcation diagrams
In general, the first three bifurcation diagrams of two parameters tend to be similar and the fourth tends to be similar to the two first ones but with some differences due to the quantization noise. The twoparameter bifurcation diagram including the quantization effects, as shown in Figure 11b, considered N = 1 and K s between 0 and 5, which satisfies the regulation conditions (steady-state error< 3 %); thus, in this area, the system can operate under stability.

CONCLUSIONS
This paper presented a detailed report of a theoretical analysis, control design and simulated and real responses of a power converter. If each process had been done separately, then the best model could not be found or the research would have been delayed. The integration of rapid control prototyping (RCP) processes with effective math tools, such as Lyapunov exponents and bifurcation diagrams, allowed the selection and validation of the most appropriate model. The paper has presented a stability analysis for the classical and two modified buck converter models controlled by ZAD and FPIC. Mathematical representations of the buck converters with the controllers were presented and the LEs were calculated. The theoretical results were compared with the simulation and experimental results by using one and two-parameter bifurcation diagrams. The quantization effects in the input variables and the signals of PWM were applied for the best model to identify the differences with the other three models. After these studies, we can conclude that the inclusion of the internal resistance r L , r s , and r M , the system becomes more stable. Thus, the test that changes the parameter K s shows that the equilibrium point is shifted to the left of the bifurcation diagram. The stability of the periodic orbit 1T for the three models of the buck converters controlled by ZAD and FPIC were determined by using LEs. When the number of bits of the measured signals and the duty cycle are decreased, the system starts losing stability, presents dynamic behaviors such as periodic bands and chaos, and becomes more chaotic. It is important to note that even with a low number of bits in the input variables and the duty cycle, the system controlled by ZAD and FPIC follows the reference signal with low error (< 3%) in the steady state.
The most elaborate model is not always the most accurate. In our illustrative case, the model that includes the effects of quantization does not achieve a better correspondence between experimental and simulated performance. We recommend gradually varying the model until a tolerable level of difference is sensed between experimental and simulated tests. The stages of modeling, control design and experimental validation should be integrated because in many cases, non-modeled dynamics could influence the calibration of the control system and the closed-loop responses.