Nonsmooth Current-Constrained Control for a DC–DC Synchronous Buck Converter with Disturbances via Finite-Time-Convergent Extended State Observers

: This study investigates the problem of overlarge current protection for a DC–DC synchronous buck converter with the existence of uncertainties and disturbances. Aiming to deal with the hardware damage in the electric circuit of a DC–DC buck that may be caused by overlarge transient current, a new nonsmooth current-constrained control (NCC) algorithm is proposed to replace the traditional ones, which use conservative coe ﬃ cients to satisfy current constraint, leading to a sacriﬁce of dynamic performance. Based on the homogeneous system technique, a nonsmooth state feedback controller is improved by adding a penalty term that prompts the adaptive gain of the controller according to the inductor current and current constraint. Then by using two ﬁnite-time extended state observers (FTESO), the unmatched disturbances and matched disturbances can be compensated to enhance the robustness of the DC–DC synchronous buck converter. The e ﬀ ect of proposed scheme has been veriﬁed by experimental results.


Introduction
Distributed power supply systems are widely used in aerospace, marine, communications, and other fields, the system only provides power bus, and the power supply inside the equipment is solved by their own power converters to improve the stability of the system and facilitate the maintenance of the system. As a kind of energy conversion device from DC to DC, a DC-DC converter has a simple structure, capable of realizing high efficiency power conversion and being modularized. DC-DC converters are widely used in power supply and load in this kind of power supply system structure [1,2].
A DC-DC converter is a kind of variable structure system with switching devices [3]. The circuit often contains capacitors, inductances and other energy storage elements, and their charging and discharging behavior has the characteristics of time-varying nonlinearity. In addition, the modeling process is too idealized, and some unmodeled dynamics are often neglected. These unmodeled dynamic characteristics are usually generated by sensors, actuators, and so on. Therefore, it is necessary to study the influence of unmodeled dynamics on a DC converter system. With the shortage of fossil energy, renewable energy technologies, such as wind energy and solar energy, have developed rapidly, and the capacity of distributed renewable energy generation systems have been increasing. However, there are many characteristics such as unpredictability, intermittency, and non-dispatch in such renewable energy systems. At the same time, there are a large number of non-linear and time-varying loads in the unmodeled dynamics, uncertainties, and external disturbances online [31]. A traditional ESO can only satisfy the requirement of asymptotic stability. Therefore, the ESO needs to be designed based on finite-time stability theory to achieve faster convergence rate and higher estimation accuracy. This paper takes the common DC-DC synchronous buck converter in a DC distributed power supply system as an example. Based on the homogeneous system technique, a nonsmooth algorithm has been designed to achieve better convergence characteristics of the DC-DC synchronous buck converter system. At the same time, a simple way to satisfy the current constraint is proposed by using barrier Lyapunonv function (BLF). To counteract the matched/unmatched disturbances, two finite-time extended state observers (FTESOs) are used which can guarantee fast convergence rate and robustness of the converter system via the super-twisting algorithm

Modeling the DC-DC Synchronous Buck Converter
The circuit topology of the DC-DC buck converter using synchronous rectification technology is shown in Figure 1, where E is the input DC voltage source, VT 1 and VT 2 are the controllable switches (VT 1 is the main switch and VT 2 is the synchronous rectifier), u ∈ [0, 1] is the duty ratio of pulse width modulation (PWM) as the control signal, V o is the output voltage, i L is the inductance current, i R is the load current, L is the filter inductor, C is the filter capacitor, and R is the load resistance. Firstly, for the buck converter with VT 1 switching on and off, the corresponding operating modes u = 1 and u = 0 respectively. requirement of asymptotic stability. Therefore, the ESO needs to be designed based on finite-time stability theory to achieve faster convergence rate and higher estimation accuracy. This paper takes the common DC-DC synchronous buck converter in a DC distributed power supply system as an example. Based on the homogeneous system technique, a nonsmooth algorithm has been designed to achieve better convergence characteristics of the DC-DC synchronous buck converter system. At the same time, a simple way to satisfy the current constraint is proposed by using barrier Lyapunonv function (BLF). To counteract the matched/unmatched disturbances, two finite-time extended state observers (FTESOs) are used which can guarantee fast convergence rate and robustness of the converter system via the super-twisting algorithm

Modeling the DC-DC Synchronous Buck Converter
The circuit topology of the DC-DC buck converter using synchronous rectification technology is shown in Figure 1, where E is the input DC voltage source, VT switching on and off, the corresponding operating modes 1 u = and 0 u = respectively. When the main switch 1 VT is on and the synchronous rectifier 2 VT is off, that is, ( ) When the main switch 1 VT is off and the synchronous rectifier 2 VT is on, that is, When the main switch VT 1 is on and the synchronous rectifier VT 2 is off, that is, u = 1.
Electronics 2020, 9, 16 4 of 14 When the main switch VT 1 is off and the synchronous rectifier VT 2 is on, that is, u = 0.
Combining Equations (1) and (2), the differential equation model of synchronous buck converter under two working modes of u = 1 and u = 0 is: The above formulas use the state space averaging method, that is to say, the final state space averaging model (Equation (3)) is obtained by averaging u = 1 and u = 0 modes over one cycle, in which V o and i L are the average values of output voltage and inductance current over a switching period [20].

Problem formulation
V o , where V r is the desired output voltage. Consider the disturbances caused by the change of load resistance and input voltage in synchronous buck converter and the uncertainties of inductance and capacitance parameters, the converter system (Equation (3)) can be rewritten as follows where x 2 , d 1 , and d 2 are denoted by and R 0 , C 0 , E 0 , and L 0 denote the nominal values of R, C, E, and L respectively.

Controller Design
Definition 1. In this article, for the convenience of writing, the following simplifications are utilized where α ∈ R, and sign( * ) is a standard sign function.
Since both matched and mismatched disturbances can lead to the decrease of the static accuracy, the first step in the design of the controller was to estimate the matched and mismatched disturbances by using two FTESOs. Secondly, a simple nonsmooth current-constrained controller based on homogeneous system theory was designed to make the output voltage follow the reference value. The control structure is shown in Figure 2. by using two FTESOs. Secondly, a simple nonsmooth current-constrained controller based on homogeneous system theory was designed to make the output voltage follow the reference value. The control structure is shown in Figure 2. Figure 2. The controller design of the synchronous buck converter.

Finite-Time Extended State Observer Design
Assumption 1 [32]. Suppose that the unknown lumped disturbances which can be describe as Equations (6) and (7) are continuously differentiable with respect to time.

Let 11
z , 21 z denote the state variable 1 x , 2 x and introduce the extended state variable 12 z , For the lumped disturbances in DC-DC synchronous buck dynamic model (Equation (4)), the FTESO proposed in [32] can be designed as follows ( ) 11 where ˆi j z is the estimation of the variables ij z , 0 ij β > is the gain of FTESO to be tuned, ( ) Assumption 2 [32,33]. The derivative of the extended state variable is unknown but bounded, i.e., existing a positive constant g such that

Finite-Time Extended State Observer Design
Assumption 1 [32]. Suppose that the unknown lumped disturbances which can be describe as Equations (6) and (7) are continuously differentiable with respect to time.
Let z 11 , z 21 denote the state variable x 1 , x 2 and introduce the extended state variable z 12 , z 22 denotes the lumped disturbance d 1 , For the lumped disturbances in DC-DC synchronous buck dynamic model (Equation (4)), the FTESO proposed in [32] can be designed as follows whereẑ ij is the estimation of the variables z ij , β ij > 0 is the gain of FTESO to be tuned, (i, j = 1, 2).
Assumption 2 [32,33]. The derivative of the extended state variable is unknown but bounded, i.e., existing a positive constant g such that g i (t) ≤ g, i = 1, 2.
By defining the observation error of the lumped disturbance as e i2 = z i2 −ẑ i2 , (i, j = 1, 2) and combining with Equations (8) and (9), we can obtain the following dynamic error equation According to [32], the dynamic error states variables in Equation (10) will converge to zero in finite time under the Assumptions 1 and 2, and the estimated valueẑ ij can converge to the real value z ij in a finite time t f , (i, j = 1, 2).
Definition 3 [18]. Consider the following nonlinear system where f : U 0 → R n is a continuous function with respect to x, and U 0 is the open neighborhood containing the origin x = 0. For a given (r 1 , · · · , r n ), if the vector function f (x) is homogeneous, then the system (Equation (13)) is homogeneous.
Lemma 1 [35]. For the following system where f (x) is a continuous vector field of homogeneous degree k < 0 with respect to (r 1 , · · · , r n ), andf (x) is a continuous vector field defined on R n . If x = 0 is the asymptotically stable equilibrium point of system . x = f (x) and satisfies for any x = 1, the following formula holds Then, x = 0 is a locally finite time equilibrium point of the system (14).
For DC-DC synchronous buck converter error dynamic equation (Equation (4)), a finite-time current-constrained controller based on the FTESO designed above is designed as whered 1 ,d 2 are the lumped disturbances estimated by FTESO, M > 0 is a constant value and 0 < γ 1 < 1, Remark 1. In this paper, considering the damage to the hardware circuit caused by current overshoot, the inductance current is limited to a certain range in the design of the controller, so that the inductance current satisfies the constraints |i L | ≤ M. It should be noted that the selection of current constraints will affect the tracking performance of the output voltage [21], so it needs to be selected appropriately according to the actual situation.
Remark 2 [21]. Unlike the BLF design method, in the backstepping algorithm [36,37], Equation (16) is to add the BLF-based non-linear term directly to the control law. When the constrained current term i L tends to the boundary value ±M, it will play a dominant role in the control law and penalizes the current, so it is also called the penalty term.

Theorem 1.
For DC-DC synchronous buck error dynamic system (Equation (4)), the designed control method (Equation (16)) can converge the output voltage to the reference set value in a finite time and satisfy the current constraint condition |i L | ≤ M if i L (0) ∈ (−M, M).
Proof. Define a candidate Lyapunov function for the system described by Equation (4) as and the first derivate of Equation (17) along Equation (4) is By substituting the controller (Equation (16)) into Equation (18), then Since the estimated value of the lumped disturbancesd 1 ,d 2 can converge to the real values d 1 , d 2 in a finite time t f , Equation (19) can be rewritten as Assume

For any initial values of current and voltage
then This indicates that Define In the same way, we can get x 2 (t), t ∈ [ 0, T) is bounded and can be expressed as Combining Equations (4)-(7) and Equation (16), it gives Since both x 1 (t) and x 2 (t) are bounded, it follows Electronics 2020, 9, 16 8 of 14 When |i L | → M, x 2 0 , we can get lim When x 2 = 0, we can get . V 2 = 0. Therefore, there exists a constant M ∈ (0, M), such that x 2 ≡ 0. Then x 1 ≡ 0 is further given by Equations (4) and (16). Based on LaSalle's invariant principle [38], it can be concluded that (x 1 (t), x 2 (t)) → 0 as t → ∞ , that is, the system (Equation (4)) is asymptotically stable under the controller (Equation (16) Under the controller (Equation (16)), the error system (Equation (4)) can be rewritten as . x where Consider the system . x and choose the Lyapunov function as then the derivative is Similar to the above, the system (Equation (31)) is asymptotically stable. Moreover, it can be verified that the system (Equation (31)) is homogeneous of degree k = (γ 1 − 1)/2 with r 1 = 1, r 2 = (γ 1 + 1)/2 by Definition 2.

Implementation and Validation
In this section, the feasibility and effectiveness of the proposed nonsmooth control algorithm was validated by using a DC-DC synchronous buck converter experimental platform. The experimental platform is shown in Figure 3, including: two DC-DC synchronous buck converters (one is used to realize the sudden change of input voltage), DSP LaunchPad TMS320F28379D (used as a control platform), a DC power supply, digital oscilloscope, DC electronic load, PC-MATLAB/Simulink (used to obtain the data from the sensors for monitoring). The synchronous buck converter in this experiment is controlled by a basic PWM gate drive, and the frequency of PWM drive signals generated by DSP is 20 kHz. Similarly, the sampling frequency of the control system is also 20 kHz. The nominal values of its parameters are listed in Table 1.   (Ω) In order to evaluate the advantages of the proposed controller, the widely used PID controller was selected for comparison. At the same time, to verify the disturbance rejection ability of the proposed method, the nonsmooth current-constrained controller was also employed in the experiment. The proper parameters of the selected controllers are listed in Table 2.  As described in Table 2, the PID controller selected a group of high-gain parameters to obtain a  In order to evaluate the advantages of the proposed controller, the widely used PID controller was selected for comparison. At the same time, to verify the disturbance rejection ability of the proposed method, the nonsmooth current-constrained controller was also employed in the experiment. The proper parameters of the selected controllers are listed in Table 2.
Case 1 (Dynamical performance under different reference voltages): In this case, the reference voltage of the synchronous buck converter changed from 15 V to 20 V at 0.1 s, and the other parameters remained the same as the nominal values. It can be seen from the output voltage and inductance current response curves in Figure 4 that the four controllers could stabilize the output voltage to the reference value. Among them, PID (High gain) had a shorter convergence time, but also had a larger transient inductor current, especially in the start-up phase, its value can reach nearly 3.8 A, which would damage the hardware circuit. Although PID (Low gain) could meet the current constraints, the convergence time of output voltage was greatly increased. Compared with PID (High gain), the dynamic performance of output voltage of the proposed control method is sacrificed to some extent to guarantee the current constrain, but it still has a short convergence time.  Figure 4 that the four controllers could stabilize the output voltage to the reference value. Among them, PID (High gain) had a shorter convergence time, but also had a larger transient inductor current, especially in the start-up phase, its value can reach nearly 3.8 A, which would damage the hardware circuit. Although PID (Low gain) could meet the current constraints, the convergence time of output voltage was greatly increased. Compared with PID (High gain), the dynamic performance of output voltage of the proposed control method is sacrificed to some extent to guarantee the current constrain, but it still has a short convergence time.  Case 2 (Robustness against sudden load resistance change): In the same way, the load resistance was reduced from 20 Ω to 10 Ω at 0.1 s by a DC electronic load. The response curves of output voltage and inductance current are shown in Figure 5. The traditional PID controllers still recovered the output voltage to 15 V after the sudden change of load resistance happened while the NCC method does not. By adding FTESOs to estimate and compensate the matched/unmatched disturbances, this problem can be solved, moreover the composite controller achieves a shorter recovery time than the PID method. Case 2 (Robustness against sudden load resistance change): In the same way, the load resistance was reduced from 20 Ω to 10 Ω at 0.1 s by a DC electronic load. The response curves of output voltage and inductance current are shown in Figure 5. The traditional PID controllers still recovered the output voltage to 15 V after the sudden change of load resistance happened while the NCC method does not. By adding FTESOs to estimate and compensate the matched/unmatched disturbances, this problem can be solved, moreover the composite controller achieves a shorter recovery time than the PID method.
was reduced from 20 Ω to 10 Ω at 0.1 s by a DC electronic load. The response curves of output voltage and inductance current are shown in Figure 5. The traditional PID controllers still recovered the output voltage to 15 V after the sudden change of load resistance happened while the NCC method does not. By adding FTESOs to estimate and compensate the matched/unmatched disturbances, this problem can be solved, moreover the composite controller achieves a shorter recovery time than the PID method.  Figure 6. It can be observed that NCC + FTESO had better  Figure 6. It can be observed that NCC + FTESO had better disturbance rejection ability and robustness compared with the NCC method and had shorter recovery time compared with the PID method. More details about the convergence time and steady state error in different cases are shown in Table 3 and Table 4.   More details about the convergence time and steady state error in different cases are shown in Tables 3 and 4.

Conclusions
In this paper, a nonsmooth current-constrained control method for a DC-DC synchronous buck converter with two finite-time extended state observers is proposed. The proposed control method used the barrier Lyapunov function to satisfy the current constraints. Then the FTESOs were used to estimate the integrated matched/unmatched disturbances and considered in the design process of the controller to achieve the better disturbance rejection ability and robustness. The feasibility of the proposed method has been verified by experimental results. Since it is still a difficult work to define a prior uncertainty bound in the actual converter system, this will be the focus of our future research [39][40][41].