A Passivity-based Control Combined with Sliding Mode Control for a DC-DC Boost Power Converter

— In this paper, a passivity-based control combined with sliding mode control for a DC-DC boost power converter is proposed. Moreover, a passivity-based control for a DC-DC boost power converter is also proposed. Using a co-ordinate transformation of state variables and control input, a DC-DC boost power converter is passive. A new plant is zero-state observable and the equilibrium point at origin of this plant is asymptotically stable. Then, a passivity-based control is applied to this plant such that the capacitor voltage is equal to the desired voltage. Additionally, the sliding mode control law is chosen such that the derivative of Lyapunov function is negative semidefinite. Finally, a passivity-based control combined with sliding mode control law is applied to this plant such that the capacitor voltage is equal to the desired voltage. The simulation results of the passivity-based control, the sliding mode control and the passivity-based control combined with sliding mode control demonstrate the effectiveness and show that the capacitor voltage is kept at the desired voltage when the desired voltage, the input voltage E and the load resistor R are changed. The results show that compared with the passivity-based control, the passivity-based control combined with sliding mode control has better performance such as shorter settling time, 8.5 ms when R changes and it has smaller steady-state error, which is indicated by the value of integral absolute error (IAE), 0.0679 when the desired voltage changes. The paper has limitations such as the assumed circuit parameters.


INTRODUCTION
The passivity-based control and its applications to power electronics are investigated by many researchers.A passivitybased control and its applications to the electromechanical applications are presented in [1], [2].[3] presented the sliding mode control and the passivity-based control.[4] presented the stability of the nonlinear systems using Lyapunov theory, a passivation and a passivity-based control of two-degree of freedom robot.Some passivity-based control approaches were presented in [5]- [8].A control system for bicycle robot based on a passivity-based method is presented in [6].
A passivity-based control of a DC-DC boost power converter using a generalized PI observer is described in [9] when the time-varying disturbance is included.Moreover, some versions of passivity-based control approaches were presented in [10]- [15].The passivity-based control of buckboost converter for different loads research was presented in [13].The passivity voltage based control of the boost power converter used in photovoltaic (PV) system was described in [16].The modified passivity-based control methods were presented in [17]- [19].[20] presented a passivity-based controller for a single-phase rectifier -DC motor system.The advantage of passivity-based control is that the equilibrium point at origin of the plant is asymptotically stable.Therefore, the capacitor voltage is convergent to the desired voltage Vd.Some nonlinear control approaches of the boost and buck converter were described in [21]- [27].[23] presented a statefeedback linearization control for output voltage regulation of a DC-DC boost converter with a constant power load.Internal model control of a DC-DC boost converter was presented in [24].[28] presented the nonlinear cascaded control for a DC-DC boost converter.
Sliding mode control can prevent the capacitor voltage from chattering and it is suitable for the switching plant such as DC-DC boost power converter.Some researches on sliding mode control were presented in [29]- [34].Estimation based sliding mode control of DC-DC boost converters was presented in [35].[36] presented fuzzy sliding mode control of DC-DC boost converter with right-half plane zero.Other control methods of dc-dc boost converter such as the adaptive control, robust control and Lyapunov theory were presented in [37]- [39].[40] presented an adaptive sliding mode control algorithm for boost DC-DC converter of FCHEV.Some improved control schemes and sliding mode control were presented in [41]- [47].A comparative analysis of conventional and sliding mode control for DC-DC boost converter was presented in [42] for PV system under transient conditions.[45] presented the cascade system control design and stability analysis for a DC-DC boost converter with proportional integral and sliding mode controller and using singular perturbation theory.Some versions of sliding mode control approaches were presented in [48]- [54].[53] presented an implementation of sliding mode voltage control controlled buck-boost converter for solar photovoltaic system.A robust sliding mode control of a DC-DC boost converter with switching frequency regulation was presented in [55].
Further, the variations of passivity-based control and its applications were presented [56]- [61].A passivity-based control using genetic algorithm for a DC-DC boost power converter is proposed in [56].[59] presented an adaptive passivity-based control of DC-DC buck power converter with constant power load in DC Microgrid systems.The passivitybased sliding mode control for the second-order nonlinear systems was presented in [60].A method to passivate a given system by using an input-output transformation matrix was described in [62].Some versions of variations of passivitybased control methods were described in [62]- [68].[65] presented the passivity-based control combining proportional integral control to improve robustness in DC Microgrids with constant power loads.[66] presented a passive backstepping control of dual active bridge converter in modular three-port DC converter.Passivity-based control combined with sliding mode control can get the advantages of both of control methods, such as the stability, small steady-state error and short settling time.
In this paper, the passivity-based control is combined with the sliding mode control to control a DC-DC boost power converter.Moreover, the passivity-based control for the same converter is also proposed.The simulation results are reported for illustration.

The contribution is
• Another approach of passivity-based control for a DC-DC boost power converter is proposed.It is different from [1], [9].
• Another approach of passivity-based sliding mode control for a DC-DC boost power converter is proposed.It is different from [1], [9], [60].The sliding mode control law is proposed such that the derivative of Lyapunov function is negative semidefinite.Then the control law is the sum of passivity-based control and sliding mode control.
The paper is organized as follows.First, the introduction is presented in section 1.The dynamical model of a DC-DC boost power converter, its passivity and the passivity-based method are presented in section 2. The design of a passivitybased control combined with the sliding mode control is described in section 3. The simulation results and discussions are described in section 4. Finally, conclusions are presented in section 5.

A. Dynamical Model of a DC-DC Boost Power Converter
A DC-DC boost power converter is described in Fig. 1.When the switch is at 2, the current  increases and stores energy in the inductor L. When the switch is at 1, the current  decreases and the energy, which is from the input voltage  and the inductor, stores in the capacitor  (and supplies in the load resistance ).The DC-DC boost power converter has the output voltage which is higher than the input voltage .The control signal is the duty ratio .In practice, the input voltage  can be the output of a rectifier or a photovoltaic system.The dynamical model [1] of the DC-DC boost power converter in Fig. 1.
Where  1 is the inductor current . 2 is the capacitor voltage . ∈ {0,1} is the switch variable (switch position).E>0 is the nominal constant value of the external voltage source. is the load resistor. is the inductor. is the capacitor.In [1], an average model of the converter is derived in the following form: Where  1 and  2 are the corresponding averaged variables. is duty ratio.The control input is  which is continuous and 0 <  < 1.The equilibrium point of ( 2) is with  = 15 (V),  = 30 (),   = 20 (V).We have  10 =0.888,  20 =20,  0 =0.25. can be the output voltage of a rectifier or a photovoltaic system and  varies. is the load resistor and  varies.  is the desired voltage of the capacitor voltage.Our goal is to regulate the capacitor voltage  at the desired value   .
Application: this circuit is used for a rectifier or a photovoltaic system.The DC load and the PV cannot connect directly and the DC-DC boost converter is needed.
The plant is passive if there exists a continuously differentiable positive semidefinite function (), which is called the storage function, such that Definition 2: Consider the plant (4) with  ≡ 0. The plant is zero-state observable if  ≡ 0 then  ≡ 0. Property [3]: Consider the plant (4).If the plant satisfies the following conditions: i) Passive with a storage function () which is positive semidefinite.

C. Passivity Property of a DC-DC Boost Power Converter
Change the variables as (5).

Stability Analysis
We have Therefore,  ̇ is negative semidefinite.

A. Passivity-based Control
According to (6), which satisfies the following conditions in property [3], the control law stabilizes the equilibrium point at origin We can choose The control law is  The plant ( 6) is passive with the positive-definite  and is zero-state observable.The function   is chosen as ( 12) is positive definite.Choose a sliding surface as ( 13) The derivative of   . ̇ =  ̇≤  ̃ ̃.For  ̇ ≤ 0, we choose the sliding mode control law as ( 14) Where  is a positive constant.

Stability analysis
=  is positive definite.

C. Passivity-based Control Combined with Sliding Mode Control
The passivity-based control combined with the sliding mode control is ( 15) Then the control law is ( 16) The control signal is the sum of the passivity-based control and the sliding mode control.

Stability Analysis
The plant ( 6) is rewritten as follows Where  ̃ is the output of the plant (6), Therefore, the function  2 is positive definite because of

A. Response to the Variations of Vd
At the beginning of the simulation, the desired voltage   is set to be 20 (V).At  = 20 ms,   is decreased to 16 (V) and at  = 40 ms,   is increased to 18 (V).At  = 60 ms,   is increased to 20 (V).

1) Passivity-based Control
The PBC results are shown in Fig. 3. Fig. 3 shows the current , the capacitor voltage  when the system is controlled by the PBC and   changes.Fig. 3 shows that at  = 20 (ms), when   is decreased to 16 (V), the capacitor voltage v has the overshoot, which is indicated by value  = |  −  2 | (V) of 1.012 V.The settling time is equal to 7.5 (ms), and  is equal to 16 (V).At  = 40 (ms), when   is increased to 18 (V), the capacitor voltage  has  (V) of 0.679 V, and  is equal to 18 (V).At  = 60 (ms), when   is increased to 20 (V), the capacitor voltage  has the value  (V) of 0.74 V, and  is equal to 20 (V).The value of IAE (integral absolute error (IAE) between   and  2 ) is 0.0688.IAE is to evaluate the performance quality of the controller.IAE is the sum of the areas below and above the desired voltage   and the voltage  The SMC results are shown in Fig. 4. Fig. 4 shows the current , the capacitor voltage  when the system is controlled by the SMC and   changes.Fig. 4 shows that at  = 20 (ms), when   is decreased to 16 (V), the capacitor voltage  has the value  (V) of 1.0818 V.The settling time is equal to 9 (ms), and  is equal to 16 (V).At  = 40 (ms), when   is increased to 18 (V), the capacitor voltage  has  (V) of 0.391 V and  is equal to 18 (V).At  = 60 (ms), when   is increased to 20 (V), the capacitor voltage  has the value  (V) of 0.535 V, and  is equal to 20 (V).The value of IAE is 0.0679.The PBC-SMC results are shown in Fig. 5. Fig. 5 shows the current , the capacitor voltage  when the system is controlled by the PBC-SMC and   changes.Fig. 5 shows that at  = 20 (ms), when   is decreased to 16 (V), the capacitor voltage  has the value  (V) of 1.0813 V.The settling time is equal to 8.5 (ms), and  is equal to 16 (V).At  = 40 (ms), when   is increased to 18 (V), the capacitor voltage  has  (V) of 0.39 V and  is equal to 18 (V).At  = 60 (ms), when   is increased to 20 (V), the capacitor voltage  has the value  (V) of 0.53 V, and  is equal to 20 (V).The value of IAE is 0.0679.When   is decreased to 16 (V), the  (V) of the proposed PBC-SMC, 1.0813 V, is larger than that of PBC, 1.012 V.When   is increased to 18 (V), the  (V) of the proposed PBC-SMC, 0.39 is smaller than that of PBC, 0.679 V.When   is increased to 20 (V), the  (V) of the proposed PBC-SMC, 0.53 V, is smaller than that of PBC, 0.74 V.The settling time of PBC, 7.5 ms, is smaller than that of PBC-SMC, 8.5 ms when   changes.The IAE of the proposed PBC-SMC, 0.0679, is smaller than that of PBC, 0.0688.Therefore, the results show that the proposed PBC-SMC provides less steady-state error when   changes and it has smaller overshoot when   is increased to 18 V and 20V.The PBC provides shorter settling time.
The results show that compared with the PBC, the proposed PBC-SMC has smaller steady-state error, which is indicated by IAE, 0.0679 when   changes and it has smaller overshoot, which is indicated by  (V), when   is increased to 18 V and 20V.However, the PBC has shorter settling time than the PBC-SMC when   changes.The results show that compared with the SMC, the PBC-SMC provides shorter settling time and smaller overshoot than the SMC.The PBC-SMC has the same IAE, 0.0679, as the SMC.The comparison results are described in Table I.
It is convenient to combine the PBC method and the SMC method because it can improve the performance, such as short settling time of the PBC and small overshoot of the SMC.

B. Response to the Variations of R
At the beginning of the simulation, the load resistor  is set to be 30 ().At  = 20 ms,  is increased to 40 () and at  = 40 ms,  is decreased to 20 ().At  = 60 ms,  is increased to 30 ().

1) Passivity-based Control
Fig. 6 is the simulation results of PBC methods when  changes.Fig. 6 shows the current , the capacitor voltage  and the resistor .Fig. 6 shows that at  = 20 (ms), when  is increased to 40 (), the capacitor voltage  has the value  (V) of 2.12 V.The settling time is equal to 8 (ms), and  is equal to 20 (V).At  = 40 (ms), when  is decreased to 20 (), the capacitor voltage  has  (V) of 5.358 V and  is equal to 20 (V).At  = 60 (ms), when  is increased to 30 (), the capacitor voltage  has the value  (V) of 4.51 V, and  is equal to 20 (V).The value of IAE is 0.0758.8 shows that at  = 20 (ms), when  is increased to 40 (), the inductor current  is equal to 0.672 (A).At  = 40 (ms), when  is decreased to 20 (), the inductor current  is equal to 1.34 (A).At  = 60 (ms), when  is increased to 30 (), the inductor current  is equal to 0.895 (A).At  = 20 (ms), when  is increased to 40 (), the capacitor voltage  has the value  (V) of 2.19 V.The settling time is equal to 8.5 (ms), and  is equal to 20 (V).At  = 40 (ms), when  is decreased to 20 (), the capacitor voltage  has  (V) of 5.4 V and  is equal to 20 (V).At  = 60 (ms), when  is increased to 30 (), the capacitor voltage  has the value  (V) of 4.5 V, and  is equal to 20 (V).The value of IAE is 0.0789.
When  is increased to 40 (), the  (V) of the proposed PBC-SMC, 2.19 V, is larger than that of PBC, 2.12 V.When  is decreased to 20 (), the  of the proposed PBC-SMC, 5.4 V, is larger than that of PBC, 5.358 V.When  is increased to 30 (), the  of the proposed PBC-SMC, 4.5 V, is smaller than that of PBC, 4.51 V.The settling time of PBC-SMC, 8.5 ms, is smaller than that of PBC, 8.8 ms or 9 ms when  changes.The IAE of the proposed PBC-SMC, 0.0789 is larger than that of PBC, 0.0758.Therefore, the results show that the proposed PBC provides less steady-state error when  changes and it has smaller overshoot when  is increased to 40  and  is decreased to 20 .The proposed PBC-SMC provides shorter settling time.
The results show that compared with PBC, the proposed PBC-SMC has shorter settling time than the PBC when  changes.However. the PBC provides smaller steady-state error when  changes and it has smaller overshoot when  is increased to 40  and  is decreased to 20 .The PBC-SMC provides shorter settling time than SMC.The PBC-SMC provides the same IAE, 0.0789, as the SMC.The comparison results are described in Table II.The PBC-SMC method can prevent the capacitor voltage  from chattering.

C. Response to the Variations of E
At the beginning of the simulation, the input voltage  is set to be 15 (V).At  = 20 ms,  is increased to 16.5 (V) and at  = 40 ms,  is decreased to 13.5 (V).At  = 60 ms,  is increased to 15 (V).

1) Passivity-based Control
Fig. 9 is the simulation results of PBC method when  changes.Fig. 9 shows the current , the capacitor voltage  and the input voltage .Fig. 9 shows that at  = 20 (ms), when  is increased to 16.5 (V), the capacitor voltage  has the value  (V) of 0.75 V.The settling time is equal to 6 (ms), and  is equal to 20 (V).At  = 40 (ms), when  is decreased to 13.5 (V), the capacitor voltage  has  (V) of 1.55 V and  is equal to 20 (V).At  = 60 (ms), when  is increased to 15 (V), the capacitor voltage  has the value  (V) of 0.95 V, and  is equal to 20 (V).The value of IAE is 0.0454.2) Sliding Mode Control Fig. 10 is the simulation results of SMC methods when  changes.Fig. 10 shows the current , the capacitor voltage  and the input voltage .Fig. 10 shows that at  = 20 (ms), when  is increased to 16.5 (V), the capacitor voltage  has the value  (V) of 0.824 V.The settling time is equal to 7 (ms) and  is equal to 20 (V).At  = 40 (ms), when  is decreased to 13.5 (V), the capacitor voltage  has  (V) of 1.47 V and  is equal to 20 (V).At  = 60 (ms), when  is increased to 15 (V), the capacitor voltage  has the value  (V) of 0.955 V and  is equal to 20 (V).The value of IAE is 0.0477.Fig. 11 is the simulation results of PBC-SMC methods when  changes.Fig. 11 shows the current , the capacitor voltage  and the input voltage .Fig. 11 shows that at  = 20 (ms), when  is increased to 16.5 (V), the inductor current  is equal to 0.815 (A).At  = 40 (ms), when  is decreased to 13.5 (V), the inductor current  is equal to 0.992 (A).At  = 60 (ms), when  is increased to 15 (V), the inductor current  is equal to 0.896 (A).At  = 20 (ms), when  is increased to 16.5 (V), the capacitor voltage  has the value  (V) of 0.82 V.The settling time is equal to 6.5 (ms), and  is equal to 20 (V).At  = 40 (ms), when  is decreased to 13.5 (V), the capacitor voltage  has  (V) of 1.45 V, and  is equal to 20 (V).At  = 60 (ms), when  is increased to 15 (V), the capacitor voltage  has the value  (V) of 0.925 V, and  is equal to 20 (V).The value of IAE is 0.0477.
When  is increased to 16.5 (V), the  (V) of the proposed PBC-SMC, 0.82 V, is larger than that of PBC, 0.75 V.When  is decreased to 13.5 (V), the  of the proposed PBC-SMC, 1.45 V, is smaller than that of PBC, 1.55 V.When  is increased to 15 (V), the  of the proposed PBC-SMC, 0.925 V, is smaller than that of PBC, 0.95 V.The settling time of PBC, 6 ms, is smaller than that of PBC-SMC, 6.5 ms or 6.7 ms when  changes.The IAE of the PBC, 0.0454, is smaller than that of PBC-SMC, 0.0477.Therefore, the results show that the PBC provides less steady-state error, and shorter settling time.The PBC-SMC provides smaller overshoot when  is decreased to 13.5 (V), and  is increased to 15 (V).
The results show that compared with the PBC-SMC, the proposed PBC has smaller steady-state error and shorter settling time when  changes.However, the proposed PBC-SMC has smaller overshoot than the PBC when  is decreased to 13.5 (V) and  is increased to 15 (V).The results show that compared with the SMC, the PBC-SMC provides shorter settling time, and smaller overshoot than the SMC.The PBC-SMC provides the same IAE, 0.0477, as the SMC.The comparison results are described in Table III.

V. CONCLUSION
In this paper, the passivity-based control combined with sliding mode control for a DC-DC boost power converter is proposed.Additionally, a standalone passivity-based control strategy for the same converter is proposed.The simulation results of the PBC, the SMC and the PBC-SMC are done with Simulink in MATLAB.The simulations results are performed in three cases of the desired voltage   changing, the input voltage variations, , and the load resistor variation, .Stability analysis of the PBC-SMC proves that the equilibrium point at origin of the plant ( 6) is asymptotically stable.Therefore, the inductor current  and the capacitor voltage  are convergent to ,   respectively.The simulation results show that the capacitor voltage  is kept at desired value   when the desired voltage   , the input voltage  and the load resistor  are changed.The results, conducted under varying conditions of   ,  and , demonstrate the effectiveness of the proposed passivitybased control and the passivity-based control combined with sliding mode control.The simulation results show that compared with the PBC-SMC, the PBC has smaller steadystate error, which is indicated by the value of IAE, 0.0758, when  changes and it has smaller overshoot indicated by when  is increased to 40  and  is decreased to 20 .However, the proposed PBC-SMC has shorter settling time, 8.5 ms than the PBC when  changes.Additionally, the results show that compared with the PBC, the proposed PBC-SMC has smaller steady-state error, which is indicated by IAE, 0.0679, when   changes and it has smaller overshoot when   is increased to 18 V and 20V.However, the PBC displays shorter settling time, 7.5 ms than the PBC-SMC when   changes.Further, the results show that compared with the PBC-SMC, the PBC provides smaller steady-state error and shorter settling time when  changes.However, it's worth noting that the proposed PBC-SMC outperforms the PBC in terms of overshoot when  is decreased to 13.5 (V) and  is increased to 15 (V).Moreover, the proposed PBC has the least value of IAE against variations of  and .The insights from our study suggest that passivity-based control combined with sliding mode control can improve the performance of DC-DC boost power converters, particularly in scenarios where quick responses to voltages variations are crucial.However, the choice of control strategy may need to be tailored to the specific requirements of the systems.The paper has limitations such as the assumed circuit parameters.Future research will explore a practical real-time experiments.The parameters of the passivity-based control are adjusted optimally.In conclusion, our study offers valuable contributions to the field of power electronics control.By demonstrating the advantages and trade-offs of different control strategies, we hope to inspire further research and innovation in the design and optimization of DC-DC boost power converters for a wide range of applications.

Fig. 2 .
Fig. 2. The structure of sliding mode control for a DC-DC boost power converter: the inductor current x1, the capacitor x2, control input  and desired voltage Vd

2 ..Fig. 3 .
Fig. 3.The results of the PBC when Vd changes: the capacitor voltage v and inductor current i 2) Sliding Mode Control

Fig. 4 .
Fig. 4. The results of the SMC when Vd changes: the capacitor voltage v and inductor current i

Fig. 5 .
Fig. 5.The results of the PBC-SMC when Vd changes: the capacitor voltage v and inductor current i

Fig. 6 .
Fig. 6.The results of the PBC when R changes: the capacitor voltage v and inductor current i and load resistor R2) Sliding Mode Control Fig.7is the simulation results of SMC methods when  changes.Fig.7shows the current , the capacitor voltage  and the resistor .

Fig. 7
Fig. 7 shows that at  = 20 (ms), when  is increased to 40 (), the capacitor voltage  has the value  (V) of 2.191 V.The settling time is equal to 10 (ms), and  is equal to 20 (V).At  = 40 (ms), when  is decreased to 20 (), the capacitor voltage  has  (V) of 5.4 V and  is equal to 20 (V).At  = 60 (ms), when  is increased to 30 (), the

Fig. 7 .
Fig. 7.The results of the SMC when R changes: the capacitor voltage v and inductor current i and load resistor R 3) Passivity-based Control Combined with Sliding Mode Control Fig. 8 is the simulation results of PBC-SMC methods when  changes.Fig. 8 shows the current , the capacitor voltage  and the resistor .

Fig. 8 .
Fig. 8.The results of the PBC-SMC when R changes: the capacitor voltage v and inductor current i and load resistor R Fig. 8 shows that at  = 20 (ms), when  is increased to 40 (), the inductor current  is equal to 0.672 (A).At  = 40 (ms), when  is decreased to 20 (), the inductor current

Fig. 9 .
Fig. 9.The results of the PBC when E changes: the capacitor voltage v and inductor current i and input voltage E

Fig. 10 .
Fig. 10.The results of the SMC when E changes: the capacitor voltage v and inductor current i and input voltage E

Fig. 11 .
Fig. 11.The results of the PBC-SMC when E changes: the capacitor voltage v and inductor current i and input voltage E

TABLE I .
THE CAPACITOR VOLTAGE V WHEN VD VARIES

TABLE II .
THE CAPACITOR VOLTAGE V WHEN R VARIES

TABLE III .
THE CAPACITOR VOLTAGE V WHEN E VARIES