6.2. Analysis of the Ideal and Non-Ideal Models in Static-State Operation
The controllability of the power converter is related to the control signal values that can be applied as a duty ratio to the converter switcher. Therefore, the operational range of the electronic device is associated with the voltage values that the converter can give at its output. This range is restricted to the interval .
The equilibrium points of the system are obtained based on the stationary state. The matrix Equation (34a) represents the equilibrium points associated with the ideal converter model [
19]. The stationary state operation of the non-linear power converter corresponds to the matrix Equation (34b).
where
,
, and
are variables referring to the stationary state around a given operational point.
The performance of the ideal and non-ideal power converters at stationary state are compared in
Figure 7. The common elements of both converters have identical characteristics. The test applies a 12 V tension at the converter input and a control signal sequence
u restricted to the range
.
The ideal converter provides an infinite voltage range, but this is impossible to achieve in real power converters due to electrical losses. These losses decrease the voltage and control input range. With a small duty ratio (0–40%), both converters have similar output voltages. However, as the duty ratio increases, the output voltage of the non-ideal converter differs significantly from the ideal.
The variation in the slope of the output voltage of the power converter must be considered for the design of the control scheme, mainly if it is based on continuous controllers. The controller cannot operate near this point because it can result in an undesirable behavior of the controlled converter. Due to the facts mentioned above, a duty cycle range of 0% to 80% is selected to ensure the stability of the controlled power converter. The selection of this range is justified because it provides better resolution for regulating the power converter. Any value outside this range is considered ineffective for the controller purpose.
6.3. Identification of the Approximated Linear Model
The analysis related to the absolute and relative grade of the power converter is required for proposing a linear model structure that can generate a dynamical response similar to the one generated by the non-linear model. As can be seen in the state–space representation associated with the power converter model developed (see Equations (27)–(29) and (33)), the absolute grade of the converter is two, comprising the capacitor voltage and the power inductor current . On the other hand, the relative grade of the system is zero because the output expression (see Equation (29)) depends explicitly on the control signal applied.
Another valuable criterion for constructing an approximated linear model from a nonlinear one is based on the analysis of the nonlinear model behavior around equilibrium points. The following matrix equations provide a state–space representation that describes the dynamics of a power converter operating at a specific operational point.
where
and
. The term
n is the number of states that characterize the system. In this case, the following change of notation is considered:
and
, then
.
The state matrix of the system (
) and the input to state vector (
) are generated by the following expressions:
where
and
. In our case, these functions can be rewritten as:
The terms
,
, and
are the state variables and the control input applied at stationary state operation. The value of
is given by the expression (33). The state-to-output vector (
) and the feed-through term (
D) are computed as follows:
where
, so:
The approximated linear model in its transfer function form can be obtained from the following equation:
The relations (34b), (37), (39) and (41) are used to generate the linear model of the system around an equilibrium point. The expression (42) is a linear structure of this power converter that models its dynamics around any equilibrium point.
where
. Also, as the power converter is stable when it operates in open loop, the real part of
is less or equal than zero. The results related to the structure of the transfer function around a set of equilibrium points are shown in
Table 2. These equilibrium points are selected inside the control signal range
.
An essential characteristic of the approximated linear models in
Table 2 is the non-minimum phase zero present in
. This behavior in a
DC-DC buck–boost power converter has been reported in prior studies, such as [
3,
18,
27]. This non-minimum phase zero restricts the operating frequencies of the converter, affecting its response time.
The recursive least-square identification algorithm with the forgetting factor is applied to identify a linear model. Analyzing the sampling time of the previously obtained results is necessary to apply the identification method. One criterion for determining the sampling time is based on the system time constant. However, this approach may not be practical in systems with oscillatory behavior. For detecting oscillations, the identification process requires a sampling time that is sufficiently small, satisfying the Nyquist/Shannon sampling period theorem, for their detection. In our study, since the equilibrium point analysis revealed a pair of complex conjugate poles, we selected the same sampling time of the simulation ( s) because it is much shorter than the oscillation period of the system, which is approximately s.
The discrete linear model, including the zero-order hold, has the following structure:
The vectors and matrix used along the identification procedure are structured as shown in the following relations:
vector (44) represents the initial values for the coefficients of the discrete model. The voltage input applied at the power input of the converter is
V. The forgetting factor chosen for the procedure equals 0.95 for obtaining robustness in the identification performance. The input and output signals used for the identification process are shown in
Figure 8a and
Figure 8b, respectively. The identification process of the values for the discrete model coefficients is shown in
Figure 8c; meanwhile, the estimation error along the whole procedure is represented in
Figure 8d.
The mean of the error associated with the identification process is equal to 8.4193 × 10
V. As shown in
Figure 8d, some error peaks represent the behavior of the algorithm to match the approximated model to the dynamics of the non-linear one.
Table 3 shows some static values of the estimated parameters. The data selected for generating the approximated linear model of the power converter are shown in the column called ”Corrected Value”. This quantity is set based on the value mean of the numeric estimation inside the variance previously computed of the data obtained.
The variance of the estimated values is always lower than its respective mean, as seen in
Table 3. This result represents that the valid values of the estimated ones never change their sign, which is essential to guarantee a suitable identification procedure.
The discrete linear model estimated is related to the transfer function in the
z-domain shown in the expression (47). This transfer function is structured considering a sampling time equivalent to
s.
This discrete model obtained in the
s-domain is equivalent to the following continuous transfer function:
The gain associated with the previous transfer function is 0.0095. This transfer function has two zeros: a minimum phase one
and a non-minimum phase one
. The poles estimated by the identification algorithm are a pair of complex conjugated poles
. The identification algorithm could detect the structure related to the converter dynamics analyzed.
Figure 9 shows the response of both models (the linear and the non-linear one) when the same control input is applied to them. Both models shown in the Figure are strongly co-related.
The frequency response of the linear model estimated is shown in
Figure 10. The initial value of the frequency response phase is 180
because of the voltage inversion of this kind of converter at its output. The gain margin is −26.7 dB, and the phase margin is −169
, so this power converter is unstable in a closed-loop configuration.
6.4. Controller Scheme Tuning
The voltage measured at the output of the power converter must be inverted to guarantee a good performance of the controller. In another way, if the reference voltage is positive, the controller can never follow that reference because of the inverting characteristic of this kind of power converter.
Figure 11 shows the frequency response of the linear model controlled by a
I controller coupled with a lead compensation action. The integral action is chosen to guarantee zero error at a stationary state. The gain associated with the controller is selected to reach an appropriate bandwidth, improving the response time. The lead compensation is commonly used for improving stability margins and achieves the desired result through the merits of its phase lead contribution [
22]. The controller designed for regulating the linear model is equivalent to the transference function shown in (49).
The phase and the gain margin of the power converter are 88.7 and 26.9 dB, respectively. This phase margin guarantees robustness at variations over the equilibrium point. The bandwidth of this device is equal to 52.3 rad/s. If this bandwidth is not greater than the commutation time associated with the switcher, then it is valid for this converter. If the application for the power converter requires a wider bandwidth, the controller must be changed to guarantee good performance. The period of the lead compensation action is s, and its alpha coefficient is .
Figure 12 shows the temporal responses of the controlled system considering both models (linear and nonlinear). The input voltage applied at the power input of the controller of the converter is 12 V.
As seen in
Figure 12, the linear controller tuned has robustness to regulate the behavior of the non-linear model of the power converter using a tuning based on the frequency analysis of a linear approximated model. The proposed methodology is practical for designing linear controllers with outstanding performance.
The result related to the maximum voltage value that the non-ideal power converter can supply justifies the use of an anti-windup configuration to avoid the unstable behavior of the controlled device, when there is a higher reference value at the controller input. It is a maximum value on the saturation control signal of
. The anti-windup gain
is defined using the following relation, and it is based on the results obtained from [
17]:
Figure 13 shows the relevance of using the anti-windup scheme, which allows for the more robust behavior of the converter controller. This figure shows how the behavior of the converter is degraded by applying a reference voltage to the controller greater than a particular value.