1. Introduction
As a kind of DC-DC power converter, the buck type has been widely used in various fields, due to its simple circuit topology, high efficiency and reliability [
1,
2]. Since buck converters generally consist of such components as the inductor, capacitor, diode and power switches (e.g., IGBT, MOSFET), they are characterized by time-varying nonlinearity. In practical applications, the traditional control approaches are mainly focused on PID [
3] and state feedback control [
4], and their control laws are generally constructed as the linear combination of the measured voltages and currents. Although these approaches are simple and easy to implement, they are sensitive to parameter perturbations and external disturbances, leading to poor robustness, and poor dynamic and static performances. For example, in [
5], the magnetic characteristics of inductors were proved to vary greatly in the presence of large magnetic flux density. In [
6], the variations of load resistor and input voltage were proved not to be neglected, especially in the case of large signal operation. Therefore, the control of buck converters with satisfactory performance is still a challenging issue to be addressed, especially in the case of parameter perturbations and external disturbances [
7,
8].
As an effective nonlinear and robust control approach, a large amount of literature has proved that sliding mode (SM) control is more suitable for the switching control of buck converters. On the one side, the SM controller can replace the traditional pulse width modulation (PWM) and realize direct ON/OFF switching control of the converters by triggering the gate of the controllable power switches. As a result, it is simple and no extra modulation circuits are needed [
9,
10]. On the other side, SM is inherently nonlinear, which differs from the approximate nonlinearity of the substitutes, like the fuzzy control [
11,
12], neural network [
13,
14] and other intelligent control approaches [
15,
16,
17]. As a result, the control performances under SM control are better; besides, the amount and time involved in the calculation can also be saved. A review of recent SM applications in power converters can be referred to [
18]. For example, in [
19], a linear sliding mode (LSM) controller was proposed to guarantee the asymptotic convergence of the output voltage; in [
20], a terminal sliding mode (TSM) control approach was proposed to improve steady accuracy and response speed by introducing finite-time convergence; in [
21], a non-singular terminal sliding mode (NTSMC) control approach was proposed to realize wide-range voltage regulation of the buck converters, due to its global finite-time convergence. Although the above SM approaches are the three commonly used types in practice, they belong to the traditional first-order SM (1-SM) [
22,
23,
24]. Attention should be paid to the chattering problem [
25], which is harmful to the stability and control performances of the system. It is closely related to switching nonlinearity, denoted as the signum function, i.e., sgn(·). In other words, when the system slides along the predesigned surface, its trajectory switches back and forth under the function of the discontinuous control law, leading to the undesired chattering phenomenon [
26,
27].
In order to solve the chattering problem, two approaches can generally be used: continuous approximation and high-order sliding mode (HOSM). For the former, a saturation function [
28] or a sigmoid-like function [
29] or the combination of SM with intelligent control [
30] can be generally introduced into a boundary layer around the sliding surface to smoothen the discontinuity of SM control law. However, this approach suffers from poor accuracy and the loss of robustness inside of the predesigned sliding surface. Meanwhile, despite the time-varying control magnitude, such intelligent control technology as the fuzzy logic [
31] and neural network [
32] depend on the experience of engineers, and the system stability cannot be guaranteed. For the latter, HOSM can be regarded as the extension of the traditional 1-SM in the sense of relative degree [
33,
34]. Its idea is to impose the switching control on the first-order derivative of the sliding variable. As a result, the chattering elimination and the expected control continuity can be achieved by the integral operation or the low-pass filtering of the switching control. At present, second-order SM (2-SM) is the most popular HOSM applied in practice. More specifically, the twisting algorithm [
35], sub-optimal algorithm [
36], and super-twisting algorithm [
37] are the often-used 2-SM. However, when facing unknown and complex disturbances, these algorithms suffer from overestimation and unnecessary control magnitudes, leading to the degradation of performances and efficiency.
In order to solve the problem of the unnecessary and fixed control gain of 2-SM, the adaptive mechanism was introduced as an effective solution to achieve time-varying control gains [
38]. At present, the often-used adaptive mechanisms can be classified into three kinds, based on the Lyapunov stability theory, intelligent control technology and the switching trajectory, respectively. For the first type, its idea is to achieve a time-varying control gain on the premise of system stability. For example, in [
39], an observer-based 2-SM approach was proposed, following the guaranteed robust stability of the observer and the controller based on the Lyapunov stability theory; in [
40,
41,
42], several adaptive 2-SM approaches were applied into uncertain or disturbing systems in practice, respectively. Compared with nominal systems, application of the adaptive mechanism is more significant. However, despite the fact that the control gain can dynamically decrease as the system converges to the equilibrium point, its relationship with the control performance is difficult to obtain. For the second type, its idea is to integrate such intelligent control technology as fuzzy logic and neural network with the 2-SM controller to achieve adaptive control magnitude, which changes as the states change [
43,
44]. However, it is heavily reliant on the prior evaluation of experts. The third type takes advantage of the inherent “switching” nature of real (non-ideal) sliding modes and its adaptive mechanism depends on counting the zero-crossing points of the sliding variable in a certain time interval [
45]. Therefore, its merits lie in the capability of online calculation, instead of traditional offline settings. Since it can adjust and maintain the control magnitude at the minimum admissible level, it is significant for engineering practice. However, some issues concerning the online zero-crossing mechanism still need to be investigated, especially for robust stability when facing the complex disturbances of buck converters.
In this paper, a novel adaptive 2-SM control approach was proposed on the basis of online zero-crossing detection to improve the control performances of buck converters with disturbances. The problems of the chattering and the fixed control gain were two key issues addressed. To be specific, the main contributions of this paper can be concluded as follows:
Without loss of generality, the possible parameter perturbations and external disturbances of the buck converters are all considered in modeling;
The robust stability of the converter system is investigated;
The adaptive mechanism of online zero-crossing detection is investigated;
The control magnitude of the controller can vary to a minimal admissible level, and the steady error of the output voltage can converge to the expected value.
This paper is organized as follows. In
Section 2, the possible parameter perturbations and external disturbances of buck converters are considered in modeling. In
Section 3, the twisting algorithm [
35] is chosen as an example of the traditional 2-SM to design a controller for the buck converter. In
Section 4, an improved twisting algorithm is proposed by combining with the adaptive mechanism of online zero-crossing detection, following its robust stability analysis. Finally, the comparative simulations, experiments and concluding remarks are given in
Section 5 and
Section 6, respectively.
5. Simulations and Experiments
In order to validate the improved twisting algorithm with the adaptive mechanism of online zero-crossing detection, as well as to show its unified robustness in converter systems with multi-disturbances, we compared it with the traditional 1-SM and the twisting algorithm, simultaneously, where the commonly used LSM was chosen as an example of 1-SM. For the convenience of analysis and comparison, they are abbreviated as “1-SMC”, “2T-SMC” and “2AT-SMC”, respectively.
For the buck converter in
Figure 1, its circuit parameters are listed in
Table 1. In the following, we comprehensively compared the three control approaches in conditions with/without disturbances by simulations and experiments.
5.1. Simulation Results and Analysis
The design of SM controllers includes two parts, i.e., a sliding surface and a control law. For 1-SMC in (6) and (7), 2T-SMC in (6) and (8), 2AT-SMC in (15), (26) and (30), the three comparative controllers were designed respectively as
where the sampling time
T = 40 μs, the given number of the zero-crossing points
N* = 8,
Λ1 = 12,
Λ2 = 24, defined in (27) and (31), respectively.
In order to compare the control performances of the above three control approaches, simulations were carried out in the following two cases, i.e., rated working condition and that with multi-disturbances.
In the case of rated working condition, the simulation comparisons of the three controllers in (38)–(40) are given in
Figure 5 and
Figure 6 and
Table 2, respectively.
When the system worked in rated working condition,
Figure 5a shows the convergence process of the sliding variable s. It can be seen that, the three controllers could all force the system to converge to the equilibrium point, but 1-SMC suffered from large oscillation amplitudes, especially at the beginning. Furthermore, we zoomed up on the local comparison diagram of 2T-SMC and 2AT-SMC. Obviously, the steady performance of 2AT-SMC was better, due to the introduction of the adaptive mechanism, where the control law
u, the output voltage
vc and inductor current
iL were the concerned performance indices for the buck converters. For the control law
u in
Figure 5b, we could see that the chattering phenomenon was serious for 1-SM, due to the switching nonlinearity of (7) [
22,
23,
24]. Since the reference value of the output voltage
vc was 5 V, the input DC voltage was 10 V, and the duty cycle of the converter was 0.5. By zooming up the local comparison of 2T-SMC and 2AT-SMC, the control signals,
u, were in the neighborhood of 0.5, due to the integral operation of 2-SM in (8), (26) and (30) [
33,
34,
38]; meanwhile, for the latter, its performance was smooth and better, which was related to the varying control gain imposed by the adaptive mechanism of the online zero-crossing detection. For the output voltage
vc and inductor current
iL in
Figure 5c,d, the corresponding performances are given in
Table 2. Obviously, the control performance of 2AT-SMC was the best, due to the improvement of the adaptive mechanism. The steady error of the output voltage
vc was 3.01 mV, the convergence time was 0.031 s; while the corresponding values of the inductor current
iL were 0.101 A and 0.032 s, which validated this paper.
In order to further test the effectiveness of the adaptive mechanism in (31), the given number of the zero-crossing points
N* was chosen as 2, 4, 8, and the comparative simulations are shown in
Figure 6. Here we took the output voltage
vc as an example, and we could see the steady errors of the three cases listed as 48.01 mV, 6.23 mV and 3.01 mV, respectively. It meant that, the smaller
N* was, the bigger the steady error was, which was in accordance with the proof in Theorem 3. It also validated the effectiveness of the proposed adaptive mechanism, based on the online zero-crossing detection [
43,
44,
45].
In practice, there generally exist multi-disturbances, and seldom is there only one type. This was also the reason that we considered the possible parameter perturbations and external disturbances of the buck converters in (5). In the following, we simulated these multi-disturbances by sine function in
Table 3. Under the control of the three kinds of SM in (38)–(40), the comparative simulations of the system with multi-disturbances are given in
Figure 7 and
Table 4, respectively.
When the buck converter system worked in disturbing condition, the simulations are given in
Figure 7 and similar conclusions could be obtained in
Figure 5 in rated working condition, i.e., the control performance of 2AT-SMC was the best, then 2T-SMC and the last was 1-SMC. Regardless of the possible parameter perturbations and external disturbances in
Table 3, the sliding variable s under the three SM approaches could converge to the equilibrium point, simultaneously, seen in
Figure 5a, due to the inherent robustness of SM. For the control law in
Figure 5b, the comparative simulations also validated the capability of 2-SM over 1-SM in chattering elimination, and the duty cycle of 2T-SMC and 2AT-SMC still maintained around 0.5, but the latter was better due to the varying control gain produced by the adaptive mechanism. The output voltage
vc and the inductor current
iL are illustrated in
Figure 5c,d, and the corresponding steady error and convergence time are given in
Table 4. By combining
Table 2 and
Table 4, the control performances of both were the best under the control of 2AT-SMC. Taking the output voltage
vc as an example, the steady errors of the buck converter in rated working condition increased from 3.01 mV to 25.41 mV in the disturbing case, while for the convergence time, it increased from 0.031 s to 0.039 s, which was due to the existence of parameter perturbations and external disturbances. Therefore, the simulations validated the proposed 2AT-SMC.
5.2. Experiment
In the following, we further tested the control performance and adaptation mechanism of the proposed 2AT-SMC by experiment. In
Figure 8a, the experiment platform, based on DSpace 1006 was given, where the sampling time was 0.5 ms and the PWM frequency 1000 Hz; and
Figure 8b is the hard ware of the buck converter.
In the experiment, we considered the possible parameter perturbations and external disturbances of the buck converters in
Table 1. Specifically, the parameter vibrations of the inductor
L and capacitor
C were in accordance with the practical elements, i.e., 0 < ∆
L < 0.1 mH and 0 < ∆
C < 10
−4 F [
46,
47]; and for the other disturbances in
Table 1, we assumed:
The input voltage E increased from 10 V to 11 V at t = 1 s, and then reduced to 10 V at t = 2 s, i.e., ∆E = 1 V;
The reference voltage Vref increased from 5 V to 5.5 V at t = 3 s, and then reduced to 5 V at t = 4 s, i.e., ∆Vref = 0.5 V;
The load resistor R increased from 10 Ω to 11 Ω at t = 5 s, and then reduced to 10Ω at t = 6 s, i.e., ∆R = 1 Ω.
The above three disturbances existed simultaneously at t = 7 s, and then disappeared at t = 8 s.
Based on the above four cases, we used the time-varying disturbances to simulate the individual, several or all disturbances in practice [
2]. The experiment results under the control of 1-SMC, 2T-SMC and 2AT-SMC are compared in
Figure 9,
Table 5 and
Table 6.
In
Table 5 and
Table 6,the performance comparisons of the steady error and response time of the output voltage
vc and inductor current
iL, respectively, are shown and they are in accordance with the time-varying information in
Figure 9. The same conclusion can be obtained in
Figure 6 and
Figure 7,
Table 2 and
Table 4 in the simulation, i.e., the control performances under the control of 2AT-SMC was the best among the three SM algorithms. Meanwhile, the three SM algorithms were all robust against the parameter perturbations and external disturbances of the buck converters. Attention should be paid to the fact that the unified robustness of 1-SMC was due to the two-value of the control
u in (7) and (38), where the detailed explanation can be seen in [
2]; while for 2T-SMC and 2AT-SMC, their unified robustness was due to the twisting algorithm itself [
35], which only needed the boundary information of these multi-disturbances. It was also the reason for the fact that the twisting algorithm is widely used in practical systems, as well as the reason that we chose it as the example of 2-SM.
Therefore, the comparative simulations and experiments in rated working condition and in disturbing condition could all validate the proposed 2AT-SMC with better robustness and control performances due to the adaptive mechanism of online zero-crossing detection. Meanwhile, it was significant for the robust control of power converters with disturbances.