Finite-Time Parameter Observer-Based Sliding Mode Control for a DC/DC Boost Converter with Constant Power Loads

: Constant power loads have negative impedance characteristics, which reduce the damping of DC/DC converter systems and have negative effects on the stability of the DC microgrid. In this paper, a ﬁnite-time parameter observer-based sliding mode controller is proposed for a boost converter with constant power loads. Firstly, a non-singular terminal sliding-mode controller is designed based on the ﬂatness of the differential and sliding mode control theory. Secondly, a ﬁnite-time observer is designed to estimate the input voltage and tunes the parameter of the controller in time. Thirdly, the ﬁnite-time stability of the closed-loop system is proved through the proposed controller. Finally, the effectiveness and robustness of the proposed controller with unknown input voltage are veriﬁed by simulation. The proposed controller can guarantee ﬁnite-time convergence without input voltage sensors, which can reduce system cost and improve system reliability.


Introduction
Energy is the material basis for promoting economic and social development. Due to the exhaustion of traditional energy and increasing air pollution, the demand for renewable energy such as water, wind, and solar energy is gradually increasing [1]. Microgrids, which can flexibly integrate energy into power systems, mainly consist of loads, energy storage devices, distributed power sources, energy conversion devices, etc. [2]. At present, renewable energy is an important power source in the microgrid [3,4]. In order to improve resource utilization and power-supply reliability, the stable operation and power dispatch of renewable energy and microgrids have become a popular research topic. In [5], a robust, optimal power-management system is proposed for hybrid AC/DC microgrids to maximize the use of renewable energy and reduce the use of fuel energy. In [6], direct sliding mode controllers were designed combined with the perturb and observe method for DC boost converters to increase the photovoltaic system's dynamic performance and ensure the stability of the system voltage.
In particular, the stable operation of DC microgrids has received a lot of attention from scholars [7,8]. Compared with AC microgrids, which have problems such as phase unbalance, harmonics, synchronization, and reactive power flow, DC microgrids are simpler to control [9,10]. In addition, DC microgrids can provide more efficient and more reliable integration of DC sources and DC loads than AC microgrids [11,12]. Nowadays, because of the increasing quantity of DC sources and the growing proportion of DC loads, DC microgrids are widely used.
DC/DC converters, which are usually used as the ports between renewable energy and the DC bus of the microgrid to ensure bus voltage stability, have become important parts of DC microgrids [13,14]. Due to the complexity of the actual working environment, DC microgrids are often affected by uncertainties such as power and load disturbances. If the DC/DC converters fail to handle these uncertainties immediately, this may cause instability was proposed, combined with feedback linearization technology. The robustness and stability analysis of the closed-loop system was clearly proved. In [44], based on feedback linearization, an SMC combined with a linear extended-state observer, which can ensure the stability of the closed-loop system with mismatched disturbance, was designed for DC/DC buck-boost converter systems. However, the above controllers can only guarantee the exponential convergence of the system. The finite-time control can put the system under a faster convergence rate and has better disturbance-rejection ability [45]. In this paper, we designed a finite-time-parameter, observer-based sliding mode controller for DC/DC boost converters with constant power loads. The specific contributions of this paper are: • A finite-time observer was designed to estimate the input voltage online and adjust the controller parameter. The observer just needs a very weak PE condition; • A finite-time-parameter, observer-based sliding mode control for the boost converter feeding a CPL was designed to ensure the finite-time stability of the closed-loop system when the input voltage is unknown; • The finite-time stability analysis of the proposed controller is also given.
The rest of the paper is arranged as follows: Section 2 describes the model of the DC/DC boost converter system. In Section 3, a finite-time controller combined with a parameter observer is designed. Section 4 shows the simulation results. Section 5 presents the conclusion. Figure 1 shows the typical circuit topology of a DC/DC boost converter with a CPL [36]. estimate the disturbance and adjust the output voltage. SMC is robust and can restrain external disturbances effectively [41,42]. Combining SMC with feedback linearization can deal with the non-linearity of the system and ensure the robustness of the system. In [43], an adaptive backstepping sliding mode control strategy for a boost converter with CPL was proposed, combined with feedback linearization technology. The robustness and stability analysis of the closed-loop system was clearly proved. In [44], based on feedback linearization, an SMC combined with a linear extended-state observer, which can ensure the stability of the closed-loop system with mismatched disturbance, was designed for DC/DC buck-boost converter systems. However, the above controllers can only guarantee the exponential convergence of the system. The finite-time control can put the system under a faster convergence rate and has better disturbance-rejection ability [45]. In this paper, we designed a finite-timeparameter, observer-based sliding mode controller for DC/DC boost converters with constant power loads. The specific contributions of this paper are:

The State Space Model of a DC/DC Boost Converter Feeding a CPL
• A finite-time observer was designed to estimate the input voltage online and adjust the controller parameter. The observer just needs a very weak PE condition; • A finite-time-parameter, observer-based sliding mode control for the boost converter feeding a CPL was designed to ensure the finite-time stability of the closed-loop system when the input voltage is unknown; • The finite-time stability analysis of the proposed controller is also given.
The rest of the paper is arranged as follows: Section 2 describes the model of the DC/DC boost converter system. In Section 3, a finite-time controller combined with a parameter observer is designed. Section 4 shows the simulation results. Section 5 presents the conclusion. Figure 1 shows the typical circuit topology of a DC/DC boost converter with a CPL [36]. The state-space averaging technique was adopted to simplify the system analysis and the controller design [46]. The mathematical model of DC/DC boost converter with CPL, which operates in continuous-conduction mode (CCM), can be written as

The State Space Model of a DC/DC Boost Converter Feeding a CPL
where vc ∈ R > 0 is the capacitance voltage or the output voltage, iL ∈ R > 0 is the inductor current, C ∈ R > 0 is the capacitance value, L ∈ R > 0 is the inductance value, P ∈ R > 0 is the power of CPL, E ∈ R > 0 is the input voltage. u = 1 − d is the control input, where d ∈ [0, 1] is the duty ratio of the switch signal. The state-space averaging technique was adopted to simplify the system analysis and the controller design [46]. The mathematical model of DC/DC boost converter with CPL, which operates in continuous-conduction mode (CCM), can be written as where v c ∈ R > 0 is the capacitance voltage or the output voltage, i L ∈ R > 0 is the inductor current, C ∈ R > 0 is the capacitance value, L ∈ R > 0 is the inductance value, P ∈ R > 0 is the power of CPL, E ∈ R > 0 is the input voltage. u = 1 − d is the control input, where d ∈ [0, 1] is the duty ratio of the switch signal.

Problem Formation
The assumption of the DC/DC boost converter with CPL is given as follows.
Assumption 1. The DC/DC boost converter feeding a CPL is working in CCM. The boost converter's parameters C, L, and P are available constants. The input voltage E is an unknown constant. The main aim of this paper is to design a control law combined with a parameter observer for System (1) to ensure the finite-time convergence of the closed-loop system. The proposed control law and the parameter observer have the following characteristics: 1.
The estimates of the parameter E can trace the real E in finite time. That iŝ

2.
The output v c of the closed-loop system can converge to the desired value v * in finite time.

The Controller Design
A finite-time-parameter, observer-based sliding mode controller is designed in this section. Firstly, through coordinate transformation, System (1) is transformed into a linear state space model. Later on, based on non-singular terminal sliding-mode theory, a finitetime controller is designed. Finally, a finite-time parameter observer is designed to estimate the unknown E and adjust the controller parameters in time.

Coordinate Transformation
In this subsection, System (1) is transformed into a linear system model by inputoutput feedback linearization. Here, we assume that the input voltage is known. The case that the input voltage is unknown is discussed later. Based on the flatness of the differential, the sum of energy in inductance and capacitance is selected as a virtual output [32]. The expressions of the virtual output and its derivative to time are as follows According to System (1), the time derivatives of the virtual output are obtained ..

y= E
It is obvious that the relative degree is equal to two for the virtual output y. Hence, inner dynamics are non-existent for this output.
The expected output voltage is v * . It is simple to determine that the expected equilibrium point is (P/E, v * ) from the dynamic model (1). Then, the expected virtual output is obtained The new states are defined that y. Let Then, the mapping from (i L , v c ) to x = (x 1 , x 2 ), which is recorded as φ, is obtained Electronics 2022, 11, 819

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The Jacobian matrix of φ is computed as While i L and v c are greater than zero, the matrix J is non-singular. Consequently, φ(i L , v c ) is a local diffeomorphism in the region where i L and v c are greater than zero. System (1) can be transformed into the following linear state space model.

The Design of the Non-singular Terminal Sliding-Mode Controller
According to sliding-mode control theory [47], a non-singular terminal sliding-mode controller is designed for System (11).
The sliding surface is designed as follows: In addition, the following control law is designed where p and q are positive odd integers and meet the condition 1 < p/q < 2, β and k are controller parameters and meet the condition β, k > 0. (11) is able to converge to zero in finite time under Sliding Surface (12) and Controller (13) for any initial state. That is

Proposition 1. The state of System
Proof of Proposition 1. The Lyapunov function is selected as follows: The expressions for the time derivative of function V 1 are obtained The time derivative of Sliding Surface (12) is derived as follows: Substituting (13) and (17) into (16), one obtains The time derivative of function V 1 is negative semi-definite. When x 2 = 0 and s = 0, we can obtain .
x 2 = 0 from Equations (11) and (13). The state x 2 and the . V 1 are not always equal to 0, when s = 0. It can be concluded from [48] that the states of System (11) with all initial conditions can reach the designed sliding surface in finite time, which is recorded as t r .
For system states in the sliding surface, we can obtain According to the model of System (11), one obtains .
The time t s from x(t r ) to x = 0 can be calculated that Then, we can obtain where t c = t r + t s . The proof of Proposition 2 is completed.

The Design of the Parameter Observer
In this subsection, we consider the case that the input voltage E is unknown. According to reference [49], a finite-time-parameter observer is designed to estimate the input voltage E online. (1), the finite-time input voltage observer, is as follows:
with the parameters λ > 0, α > 0. w c (t) is defined as where ξ ∈ (0, 1) is a constant chosen by designer. We assume that ∃ t e > 0 satisfies the following inequality: then the observer outputÊ can trace the real E in finite time. That is, Proof of Proposition 2. According to System (1), we can obtain (24) and (25)), where n := d dt . The error The solution of the above function is Substituting the solution of (27) into (33), it is obtained that From Equation (29) and Inequality (30), we can obtain Therefore, according to Equations (35) and (36), it can be obtained that Under Function (28), we haveÊ The proof of the observer-estimation finite-time convergence is completed. Through replacing the unknown E by the output of the parameter observerÊ, a finite-time-parameter, observer-based sliding mode controller is obtained (38) Figure 2 shows the structure of the proposed controller. Through replacing the unknown E by the output of the parameter observer E , a finite-time-parameter, observer-based sliding mode controller is obtained Figure 2 shows the structure of the proposed controller.

Stability Analysis
Proposition 3. Under Controller (38) with Observer (23)- (28), the output of System (1) can converge to the reference output voltage v * in finite time.
Proof of Proposition 3. According to Proposition 3, when t > t e , E = E. Consequently, we divide the proof process into two steps. Firstly, we prove that the states , , , are bounded in the finite time t e . Secondly, we prove that the bounded states can converge to the expected states under Controller (38) with the condition E = E in finite time.
Step 1: A function is defined that The time derivative of is as follows: (1)and (11) ,

Stability Analysis
Proposition 3. Under Controller (38) with Observer (23)- (28), the output of System (1) can converge to the reference output voltage v * in finite time.
Proof of Proposition 3. According to Proposition 3, when t > t e ,Ê = E. Consequently, we divide the proof process into two steps. Firstly, we prove that the states x 1 , x 2 , i L , v c are bounded in the finite time t e . Secondly, we prove that the bounded states can converge to the expected states under Controller (38) with the conditionÊ = E in finite time.
Step 1: A function is defined that The time derivative of V 2 is as follows: . (1) and (11)), (38)), Since K V and L V are constants, the function V 2 is bounded, when t ≤ t e . It can be concluded from [50] that x 1 , x 2 , i L , v c are bounded and cannot escape into infinity in the finite time t e .
Step 2: when t > t e ,Ê = E. We can obtain , v c (t e ) are considered as the initial time and the finite initial, respectively. We consider the stability of System (11) under Sliding Surface (42) and Control Law (43). According to Proposition 2, the state of System (11) is able to converge to zero in finite time. That is,

Remark 1.
The proposed controller can ensure the finite-time convergence of the closed-loop system in the case that the input voltage is unknown. Compared with the exponential convergence, the finitetime convergence can make the system have better transient performance and stronger robustness. Moreover, the controller-design step can be extended to the design of controllers for DC/DC boost converters with resistive loads, constant current loads, or mixed loads.

Remark 2.
We did not apply any treatments for chattering in this paper. Currently, many methods are available to restrain chattering. By using a disturbance observer, chattering can be weakened. The switching function gain k can be reduced to the upper bound of the observation error, which can realize the suppression of chattering [44,51].

Simulation Results
In this section, the proposed controller is used to control the boost converter feeding a CPL in a MATLAB/Simulink environment, and its effectiveness is verified. The simulation schematic of the proposed controller is shown in Figure 3. From the reference [33], the parameters adopted by boost converter are shown in Table 1. The initial output of the proposed observer is selected as 9 V to imitate the case that the input voltage E is unknown. The sliding surface parameters p and q are chosen as 5 and 3, respectively.

Simulation Results
In this section, the proposed controller is used to control the boost converter feeding a CPL in a MATLAB/Simulink environment, and its effectiveness is verified. The simulation schematic of the proposed controller is shown in Figure 3. From the reference [33], the parameters adopted by boost converter are shown in Table 1. The initial output of the proposed observer is selected as 9 V to imitate the case that the input voltage E is unknown. The sliding surface parameters p and q are chosen as 5 and 3, respectively.  Firstly, we test the performance of the proposed control method. The reference output voltage is set as v * = 40 V. The observer parameters are set as λ = 10, α = 0.00005. Figure 4 shows the transient of the closed-loop system. We can improve the convergence speed by increasing the values of k/β and β appropriately. However, a large value of k causes strong chattering and a large value of β causes numerical errors in practice. When we select the values of k and β, we should consider keeping the balance between the transient performance of the system and the amplitude of the chattering, and also need to consider the feasibility in practice. Figure 5 shows the transients of the controller with different parameters for the user to choose and reference according to the actual situation.  Firstly, we test the performance of the proposed control method. The reference output voltage is set as v * = 40 V. The observer parameters are set as λ = 10, α = 0.00005. Figure 4 shows the transient of the closed-loop system. We can improve the convergence speed by increasing the values of k/β and β appropriately. However, a large value of k causes strong chattering and a large value of β causes numerical errors in practice. When we select the values of k and β, we should consider keeping the balance between the transient performance of the system and the amplitude of the chattering, and also need to consider the feasibility in practice. Figure 5 shows the transients of the controller with different parameters for the user to choose and reference according to the actual situation.       Moreover, we consider the step change in reference output voltage v * . At 0.02 s, v * is changed from 40 V to 50 V. When t = 0.04 s, v * is increased to 60 V.
The simulation result is revealed in Figure 6. The output voltage can track reference voltage precisely and quickly. Moreover, we consider the step change in reference output voltage v * . At 0.02 s, v * is changed from 40 V to 50 V. When t = 0.04 s, v * is increased to 60 V.
The simulation result is revealed in Figure 6. The output voltage can track reference voltage precisely and quickly.

Scenario 2
Here, we test the tracking performance of the presented parameter observer with the controller's parameters k = 800,000, β = 400,000. When t = 0.05 s, there is a step change in input voltage E.
The output of the observer with different and are shown in Figure 7. We can increase the values of λ and α appropriately to improve the tracking performance. The voltage estimates can converge to the real input voltage in finite time. After 0.05 s, the estimates still track the actual voltage in a short amount of time.

Scenario 3
Finally, we verify the stability of the closed-loop system when there is a step change in the input voltage E. It changes from 15 V to 20 V at 0.04 s. The controller parameters are k = 1,000,000, β = 500,000, λ = 60 and α = 0.000005. Figure 8 demonstrates the simulation results. At 0.05 s, the output voltage converges to the reference voltage rapidly only after a small fluctuation. The proposed control method can effectively reduce the impact of input voltage disturbance on the system.

Scenario 2
Here, we test the tracking performance of the presented parameter observer with the controller's parameters k = 800, 000, β = 400, 000. When t = 0.05 s, there is a step change in input voltage E.
The output of the observer with different λ and α are shown in Figure 7. We can increase the values of λ and α appropriately to improve the tracking performance. The voltage estimates can converge to the real input voltage in finite time. After 0.05 s, the estimates still track the actual voltage in a short amount of time. Moreover, we consider the step change in reference output voltage v * . At 0.02 s, v * is changed from 40 V to 50 V. When t = 0.04 s, v * is increased to 60 V.
The simulation result is revealed in Figure 6. The output voltage can track reference voltage precisely and quickly.

Scenario 2
Here, we test the tracking performance of the presented parameter observer with the controller's parameters k = 800,000, β = 400,000. When t = 0.05 s, there is a step change in input voltage E.
The output of the observer with different and are shown in Figure 7. We can increase the values of λ and α appropriately to improve the tracking performance. The voltage estimates can converge to the real input voltage in finite time. After 0.05 s, the estimates still track the actual voltage in a short amount of time.

Scenario 3
Finally, we verify the stability of the closed-loop system when there is a step change in the input voltage E. It changes from 15 V to 20 V at 0.04 s. The controller parameters are k = 1,000,000, β = 500,000, λ = 60 and α = 0.000005. Figure 8 demonstrates the simulation results. At 0.05 s, the output voltage converges to the reference voltage rapidly only after a small fluctuation. The proposed control method can effectively reduce the impact of input voltage disturbance on the system.

Scenario 3
Finally, we verify the stability of the closed-loop system when there is a step change in the input voltage E. It changes from 15 V to 20 V at 0.04 s. The controller parameters are k = 1, 000, 000, β = 500, 000, λ = 60 and α = 0.000005. Figure 8 demonstrates the simulation results. At 0.05 s, the output voltage converges to the reference voltage rapidly only after a small fluctuation. The proposed control method can effectively reduce the impact of input voltage disturbance on the system.

Conclusions
In this paper, we solved the finite-time control problem of a DC/DC boost converter feeding a CPL in the case that input voltage is unknown. Firstly, the non-linear state-space average model of the boost converter feeding a constant power load is transformed into a linear state-space model based on the flatness of the differential, which can address the non-linear and non-minimum phase problems of the system. Secondly, a non-singular terminal sliding-mode controller was designed for the linear system to ensure the finitetime stability and robustness of the system. Thirdly, in order to estimate the input voltage, a finite-time input voltage observer was designed. The output of the observer can trace the real input voltage E in finite time, which can improve the ability of the system to suppress disturbances caused by input-voltage fluctuations. By replacing the parameter E of the above non-singular terminal sliding-mode controller with the output of the observer, a finite-time-parameter, observer-based sliding mode controller was developed. Finally, simulation results verified the effectiveness of the designed control method. The proposed controller can ensure the finite-time stability of the closed-loop system without input voltage information. Compared with existing control techniques, which can ensure exponential stability, the controller proposed in this paper can provide faster

Conclusions
In this paper, we solved the finite-time control problem of a DC/DC boost converter feeding a CPL in the case that input voltage is unknown. Firstly, the non-linear state-space average model of the boost converter feeding a constant power load is transformed into a linear state-space model based on the flatness of the differential, which can address the non-linear and non-minimum phase problems of the system. Secondly, a non-singular terminal sliding-mode controller was designed for the linear system to ensure the finitetime stability and robustness of the system. Thirdly, in order to estimate the input voltage, a finite-time input voltage observer was designed. The output of the observer can trace the real input voltage E in finite time, which can improve the ability of the system to suppress disturbances caused by input-voltage fluctuations. By replacing the parameter E of the above non-singular terminal sliding-mode controller with the output of the observer, a finite-time-parameter, observer-based sliding mode controller was developed. Finally, simulation results verified the effectiveness of the designed control method. The proposed controller can ensure the finite-time stability of the closed-loop system without input voltage information. Compared with existing control techniques, which can ensure exponential stability, the controller proposed in this paper can provide faster convergence, higher tracking accuracy, and greater robustness, while the use of input-voltage sensors can be reduced to lower the cost of the system.
In future work, we plan to design an observer for the DC/DC converters with CPLs to realize the simultaneous estimation of inductance current and uncertain parameters. Based on the above ideas, an adaptive sensorless controller for converters is under study.