Maximum Power Extraction of Photovoltaic Systems Using Dynamic Sliding Mode Control and Sliding Observer

Abstract

In this paper, a robust optimized controller is implemented in the photovoltaic generator system (PVGS). The PVGS is composed of individual photovoltaic (PV) cells, which convert solar energy to electrical energy. To optimize the efficiency of the PVGS under variable solar irradiance and temperatures, a maximum power point tracking (MPPT) controller is necessary. Additionally, the PVGS output voltage is typically low for many applications. To achieve the MPPT and to gain the output voltage, an increasing boost converter (IBC) is employed. Then, two issues should be considered in MPPT. At first, a smooth control signal for adjusting the duty cycle of the IBC is important. Another critical issue is the PVGS and IBC unknown sections, i.e., the total system uncertainty. Therefore, to address the system uncertainties and to regulate the smooth duty cycle of the converter, a robust dynamic sliding mode control (DSMC) is proposed. In DSMC, a low-pass integrator is placed before the system to suppress chattering and to produce a smooth actuator signal. However, this integrator increases the system states, and hence, a sliding mode observer (SMO) is proposed to estimate this additional state. The stability of the proposed control scheme is demonstrated using the Lyapunov theory. Finally, to demonstrate the effectiveness of the proposed method and provide a reliable comparison, conventional sliding mode control (CSMC) with the same proposed SMO is also implemented.

1. Introduction

The increase in the use of fossil fuels leads to global warming. This has driven significant adoption of wind turbines [1,2] or solar energy [3,4], because they are accessible, renewable, and clean [5]. In particular, solar energy has gained attention due to the cost-effectiveness of PVGS worldwide [6]. However, the output power of PVGS depends on solar irradiation and ambient temperature [7], which are not constant, i.e., are variable and fluctuate over time [6,7]. To improve and enhance the PVGS efficiency, it is essential to use an MPPT controller [8,9].

Typically, MPPT is achieved through the duty cycle adjusting of power converter following the PVGS [4,8], often using an IBC. Various MPPT controllers have been proposed in the literature [9], such as linear [10,11] or nonlinear approaches [12,13,14,15,16,17,18,19,20].

For example, in [10], the settling time and overshoot are reduced through a PID controller. Another linear strategy is based on LQR [11] for IBC. But linear controllers are not suitable for MPPT [9]. The P&O method is known as a nonlinear procedure, which can oscillate near the maximum power [12,13] or can remain in a local maximum [12,13]. In [13], a combination of adaptive PID and P&O was used. The PID is used to improve the performance of P&O.

Another nonlinear approach including PSO [14], model predictive controller [15], neural networks [16], Chaotic Horse Herd Optimizer [17], fuzzy controllers [18], ANFIS, and fuzzy sliding mode controller [19,20]. Note that all these intelligent approaches might not be very representative, as they suffer from computational cost. Among all of them, just sliding mode control is invariant [20].

Due to the available uncertainties of the total system in PVGS and IBC, a robust approach such as SMC is essential to regulate the duty cycle of the converter. SMC is known for the invariant as its important property, which is stronger than the robustness [21,22,23]. Hence, SMC is used in many MPPT controllers [24,25,26,27,28,29,30]. However, many SMC-based MPPT controllers suffer from chattering due to the use of the Signum function [20,21,22], which prevents a smooth duty cycle.

To eliminate chattering, several methods have been proposed [31], such as BL-SMC [32], ABL-SMC [33,34,35], HOSMC [36,37,38], and DSMC [39,40,41]. However, the first two BL-SMC and ABL-SMC lose the invariance property [31,32]. Because in these methods, the system states are provided into the defined layer. HOSMC uses an observer to estimate the higher-order derivatives of the system model [42,43,44].

In the last approach, i.e., DSMC, the discontinuous Signum function effect is mitigated by a low-pass integrator filter before the system, increasing the system states that need to be estimated by an observer [39,40,41]. Thus, in DSMC, the increased state should be estimated, while in HOSMC, the system model differentiation must be estimated, making DSMC preferable.

The concepts like model identifications [45,46], parameter identifications [47,48], and disturbance observers [49] are used in controlling systems (linear or nonlinear), utilizing system outputs as feedback. Additionally, SMO is invariant, similar to SMC. Therefore, SMO maintains invariance against disturbance or uncertainty [50,51].

Motivated by the critical issue of chattering, the DSMC has been proposed in this paper to have MPPT of PVGS and preserve invariance to address model uncertainty, and also reliably prevent chattering. Then, a new SMO is constructed to implement this controller and to identify the system model. The Lyapunov theory is then implemented to prove the stabilization of the DSMC and SMO.

Therefore, the overall contribution of this paper can be summarized as follows:

• A new MPPT approach is implemented for PVGS.
• A new structure of robust DSMC is proposed.
• The chattering is removed using DSMC.
• A smooth input control is obtained for the duty cycle of IBC.
• A new scheme of SMO is proposed to estimate the unknown uncertain of the total system (PVGS and IBC).

The proposed approach is detailed in six sections. Section 2 presents the photovoltaic model structure. Section 3 explains the MPPT model and linear state feedback. Section 4 discusses the proposed SMO and DSMC. Finally, Section 5 and Section 6 provide simulation results and conclusions, respectively.

Moreover,

Table 1. Nomenclature.

Parameter	Acronym
Adaptive Boundary Layer Sliding Mode Control	ABL-SMC
Adaptive Neuro-Fuzzy Inference System	ANFIS
Boundary Layer Sliding Mode Control	BL-SMC
Conventional Sliding Mode Control	CSMC
Dynamic Sliding Mode Control	DSMC
Higher Order Sliding Mode Control	HOSMC
Increasing Boost Converter	IBC
Linear Quadratic Regulator	LQR
Maximum Power Point Tracking	MPPT
Particle Swarm Optimization	PSO
Perturb and Observed	P&O
Photovoltaic	PV
Photovoltaic Generator Systems	PVGS
Proportional Integral Derivative	PID
Pulse Width Modulation	PWM
Root Mean Square	RMS
Sliding Mode Control	SMC
Sliding Mode Observers	SMO
Symmetric Positive Definite	SPD

shows all the acronyms that are used in this paper.

2. Photovoltaic Model Structure

The PVGS has two inputs: $\mathrm{\lambda }$ as the solar irradiation and $\mathrm{T}$ as the cell temperature. Then, the complete series system, which includes the PVGS and the DC-DC IBC, is illustrated in Figure 1 [4,27].

In this figure, $\mathrm{i}(\mathrm{t})$ and $\mathrm{v}(\mathrm{t})$ are PVGS current and voltage outputs, ${\mathrm{I}}_{\mathrm{c}}$ and ${\mathrm{V}}_{\mathrm{c}}$ are the fixed current and voltage outputs of the IBC, and ${\mathrm{T}}_{\mathrm{d}}$ is the duty cycle of the IBC (the output of PWM).

2.1. PVGS Model

The output current of a single PV cell, depicted in Figure 2, is given by the following equation [8,9,10,11,12,13,14,15,16,17,18,19,23,24,25,26,27,28,29,30].

Figure 2. A single PV cell.

Figure 2. A single PV cell.

$${\mathrm{I}}_{\mathrm{P}\mathrm{V}}={\mathrm{I}}_{\mathrm{p}\mathrm{h}}-{\mathrm{I}}_{\mathrm{D}}-{\mathrm{I}}_{\mathrm{s}\mathrm{h}}$$

(1)

where ${\mathrm{I}}_{\mathrm{p}\mathrm{h}}$, ${\mathrm{I}}_{\mathrm{D}}$, and ${\mathrm{I}}_{\mathrm{s}\mathrm{h}}$ represent the current produced by incident light, diode current, and parallel current, respectively.

$${\mathrm{I}}_{\mathrm{p}\mathrm{h}}=\left[{\mathrm{I}}_{\mathrm{s}\mathrm{c}}+{\mathrm{K}}_{\mathrm{I}}(\mathrm{T}-{\mathrm{T}}_{\mathrm{r}})\right]{\displaystyle \frac{\mathrm{\lambda }}{1000}}$$

(2)

$${\mathrm{I}}_{\mathrm{D}}={\mathrm{I}}_{\mathrm{r}}\left[\exp \left({\displaystyle \frac{{\mathrm{V}}_{\mathrm{P}\mathrm{V}}+{\mathrm{R}}_{\mathrm{s}}{\mathrm{I}}_{\mathrm{P}\mathrm{V}}}{{\mathrm{V}}_{\mathrm{T}}}}\right)-1\right]$$

(3)

$${\mathrm{I}}_{\mathrm{s}\mathrm{h}}={\displaystyle \frac{{\mathrm{V}}_{\mathrm{P}\mathrm{V}}+{\mathrm{R}}_{\mathrm{s}}{\mathrm{I}}_{\mathrm{P}\mathrm{V}}}{{\mathrm{R}}_{\mathrm{s}\mathrm{h}}}}$$

(4)

Using $\mathrm{q}=1.60217\times {10}^{-19} \mathrm{C}$ as the electron charge and $\mathrm{k}=1.38\times {10}^{-23} \mathrm{J}/\mathrm{K}$ as the Boltzmann constant, the thermal voltage is defined as ${\mathrm{V}}_{\mathrm{T}}={\displaystyle \frac{\mathrm{k}\mathrm{T}}{\mathrm{q}}}$. Moreover, ${\mathrm{I}}_{\mathrm{r}}$ is cell reverse saturation current, ${\mathrm{T}}_{\mathrm{r}}$ is reference temperature, ${\mathrm{I}}_{\mathrm{s}\mathrm{c}}$ and ${\mathrm{K}}_{\mathrm{I}}$ are the short-circuit current and temperature coefficient in the short-circuit case, both at reference condition, and ${\mathrm{I}}_{\mathrm{P}\mathrm{V}}$ and ${\mathrm{V}}_{\mathrm{P}\mathrm{V}}$ are the output current and voltage of the PV. Using ${\mathrm{N}}_{\mathrm{p}}$ and ${\mathrm{N}}_{\mathrm{s}}$ as the numbers of parallel and series arrays of PV cells in a PVGS, the relationship between its output current, i.e., $\mathrm{i}(\mathrm{t})$ and its output voltage, i.e., $\mathrm{v}(\mathrm{t})$, is as follows:

$$\mathrm{i}(\mathrm{t})={\mathrm{I}}_{\mathrm{p}\mathrm{h}\mathrm{g}}-{\mathrm{I}}_{\mathrm{r}\mathrm{g}}\left[\exp \left({\displaystyle \frac{\mathrm{v}(\mathrm{t})+{\mathrm{R}}_{\mathrm{s}\mathrm{g}}\mathrm{i}(\mathrm{t})}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}}\right)-1\right]-{\displaystyle \frac{\mathrm{v}(\mathrm{t})+{\mathrm{R}}_{\mathrm{s}\mathrm{g}}\mathrm{i}(\mathrm{t})}{{\mathrm{R}}_{\mathrm{s}\mathrm{h}\mathrm{g}}}}$$

(5)

such that ${\mathrm{V}}_{\mathrm{T}\mathrm{g}}={\displaystyle \frac{{\mathrm{N}}_{\mathrm{s}}}{{\mathrm{N}}_{\mathrm{p}}}}{\mathrm{V}}_{\mathrm{T}}$, ${\mathrm{R}}_{\mathrm{s}\mathrm{g}}={\displaystyle \frac{{\mathrm{N}}_{\mathrm{s}}}{{\mathrm{N}}_{\mathrm{p}}}}{\mathrm{R}}_{\mathrm{s}}$, ${\mathrm{R}}_{\mathrm{s}\mathrm{h}\mathrm{g}}={\displaystyle \frac{{\mathrm{N}}_{\mathrm{s}}}{{\mathrm{N}}_{\mathrm{p}}}}{\mathrm{R}}_{\mathrm{s}\mathrm{h}}$, ${\mathrm{I}}_{\mathrm{r}\mathrm{g}}={\mathrm{N}}_{\mathrm{p}}{\mathrm{I}}_{\mathrm{r}}$, and ${\mathrm{I}}_{\mathrm{p}\mathrm{h}\mathrm{g}}={\mathrm{N}}_{\mathrm{p}}{\mathrm{I}}_{\mathrm{p}\mathrm{h}}$.

2.2. IBC Model

Figure 3 shows an IBC, which is described by the following equations, where ${\mathrm{i}}_{\mathrm{l}\mathrm{i}}$ is the inductor current.

Figure 3. The IBC structure.

Figure 3. The IBC structure.

$$\begin{array}{l}{\mathrm{C}}_{\mathrm{i}}{\displaystyle \frac{\mathrm{d}\mathrm{v}(\mathrm{t})}{\mathrm{d}\mathrm{t}}}=\mathrm{i}(\mathrm{t})-{\mathrm{i}}_{\mathrm{l}\mathrm{i}}\\ {\mathrm{L}}_{\mathrm{i}}{\displaystyle \frac{\mathrm{d}{\mathrm{i}}_{\mathrm{l}\mathrm{i}}}{\mathrm{d}\mathrm{t}}}=\mathrm{v}(\mathrm{t})-(1-{\mathrm{T}}_{\mathrm{d}}){\mathrm{V}}_{\mathrm{c}}\end{array}$$

(6)

In which, ${\mathrm{C}}_{\mathrm{i}}$ and ${\mathrm{L}}_{\mathrm{i}}$ are input capacitor and self-inductor, respectively, ${\mathrm{C}}_{\mathrm{o}}$ is the output capacitor, and $\mathrm{Q}$ is the semiconductor switch. The dynamic of ${\mathrm{C}}_{\mathrm{o}}$ is ignored because it is supposed the output voltage ${\mathrm{V}}_{\mathrm{c}}$ and output current ${\mathrm{I}}_{\mathrm{c}}$ are constant. In addition, duty cycle ${\mathrm{T}}_{\mathrm{d}}$ is defined in Figure 1.

3. MPPT Structure

The MPPT of PVGS is achieved if the derivative of output power with respect to voltage is zero. Then, the state variable $\mathrm{x}$ is defined as follows and should be zero:

$$\mathrm{x}={\displaystyle \frac{\partial \mathrm{p}}{\partial \mathrm{v}}}={\displaystyle \frac{\partial (\mathrm{v}\mathrm{i})}{\partial \mathrm{v}}}=\mathrm{i}+\mathrm{v}{\displaystyle \frac{\partial \mathrm{i}}{\mathrm{d}\mathrm{v}}}$$

(7)

Now we have

$$\begin{array}{ll}\dot{\mathrm{x}}& ={\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{t}}}+{\displaystyle \frac{\partial \mathrm{v}}{\partial \mathrm{t}}}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}+\mathrm{v}{\displaystyle \frac{\partial }{\partial \mathrm{t}}}\left({\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}\right)={\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{t}}}+{\displaystyle \frac{\partial \mathrm{v}}{\partial \mathrm{t}}}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}+\mathrm{v}{\displaystyle \frac{\partial }{\partial \mathrm{v}}}\left({\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{t}}}\right)={\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}{\displaystyle \frac{\partial \mathrm{v}}{\partial \mathrm{t}}}+{\displaystyle \frac{\partial \mathrm{v}}{\partial \mathrm{t}}}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}+\mathrm{v}{\displaystyle \frac{\partial }{\partial \mathrm{v}}}\left({\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}{\displaystyle \frac{\partial \mathrm{v}}{\partial \mathrm{t}}}\right)\\ & =2\dot{\mathrm{v}}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}+\mathrm{v}{\displaystyle \frac{\partial }{\partial \mathrm{v}}}\left(\dot{\mathrm{v}}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}\right)=2\dot{\mathrm{v}}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}+\mathrm{v}\left({\displaystyle \frac{\partial \dot{\mathrm{v}}}{\partial \mathrm{v}}}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}+\dot{\mathrm{v}}{\displaystyle \frac{{\partial }^2\mathrm{i}}{\partial {\mathrm{v}}^2}}\right)=2\dot{\mathrm{v}}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}+\mathrm{v}\dot{\mathrm{v}}{\displaystyle \frac{{\partial }^2\mathrm{i}}{\partial {\mathrm{v}}^2}}+\mathrm{v}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}{\displaystyle \frac{\partial \dot{\mathrm{v}}}{\partial \mathrm{v}}}\end{array}$$

(8)

and also

$$\begin{array}{ll}\ddot{\mathrm{x}}& =2\ddot{\mathrm{v}}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}+2\dot{\mathrm{v}}{\displaystyle \frac{\partial }{\partial \mathrm{t}}}\left({\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}\right)+{\dot{\mathrm{v}}}^2{\displaystyle \frac{{\partial }^2\mathrm{i}}{\partial {\mathrm{v}}^2}}+\mathrm{v}\ddot{\mathrm{v}}{\displaystyle \frac{{\partial }^2\mathrm{i}}{\partial {\mathrm{v}}^2}}+\mathrm{v}\dot{\mathrm{v}}{\displaystyle \frac{\partial }{\partial \mathrm{t}}}\left({\displaystyle \frac{{\partial }^2\mathrm{i}}{\partial {\mathrm{v}}^2}}\right)+{\displaystyle \frac{\partial }{\partial \mathrm{t}}}\left(\mathrm{v}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}{\displaystyle \frac{\partial \dot{\mathrm{v}}}{\partial \mathrm{v}}}\right)\\ & =2\ddot{\mathrm{v}}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}+2\dot{\mathrm{v}}{\displaystyle \frac{\partial }{\partial \mathrm{v}}}\left({\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{t}}}\right)+{\dot{\mathrm{v}}}^2{\displaystyle \frac{{\partial }^2\mathrm{i}}{\partial {\mathrm{v}}^2}}+\mathrm{v}\ddot{\mathrm{v}}{\displaystyle \frac{{\partial }^2\mathrm{i}}{\partial {\mathrm{v}}^2}}+\mathrm{v}\dot{\mathrm{v}}{\displaystyle \frac{{\partial }^2}{\partial {\mathrm{v}}^2}}\left({\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{t}}}\right)+{\displaystyle \frac{\partial }{\partial \mathrm{t}}}\left(\mathrm{v}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}{\displaystyle \frac{\partial \dot{\mathrm{v}}}{\partial \mathrm{v}}}\right)\\ & =2\ddot{\mathrm{v}}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}+2\dot{\mathrm{v}}{\displaystyle \frac{\partial }{\partial \mathrm{v}}}\left(\dot{\mathrm{v}}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}\right)+{\dot{\mathrm{v}}}^2{\displaystyle \frac{{\partial }^2\mathrm{i}}{\partial {\mathrm{v}}^2}}+\mathrm{v}\ddot{\mathrm{v}}{\displaystyle \frac{{\partial }^2\mathrm{i}}{\partial {\mathrm{v}}^2}}+\mathrm{v}\dot{\mathrm{v}}{\displaystyle \frac{{\partial }^2}{\partial {\mathrm{v}}^2}}\left(\dot{\mathrm{v}}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}\right)+{\displaystyle \frac{\partial }{\partial \mathrm{t}}}\left(\mathrm{v}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}{\displaystyle \frac{\partial \dot{\mathrm{v}}}{\partial \mathrm{v}}}\right)\\ & =2\ddot{\mathrm{v}}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}+2\dot{\mathrm{v}}\left({\displaystyle \frac{\partial \dot{\mathrm{v}}}{\partial \mathrm{v}}}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}+\dot{\mathrm{v}}{\displaystyle \frac{{\partial }^2\mathrm{i}}{\partial {\mathrm{v}}^2}}\right)+{\dot{\mathrm{v}}}^2{\displaystyle \frac{{\partial }^2\mathrm{i}}{\partial {\mathrm{v}}^2}}+\mathrm{v}\ddot{\mathrm{v}}{\displaystyle \frac{{\partial }^2\mathrm{i}}{\partial {\mathrm{v}}^2}}+\mathrm{v}\dot{\mathrm{v}}\left({\displaystyle \frac{{\partial }^2\dot{\mathrm{v}}}{\partial {\mathrm{v}}^2}}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}+\dot{\mathrm{v}}{\displaystyle \frac{{\partial }^3\mathrm{i}}{\partial {\mathrm{v}}^3}}\right)+{\displaystyle \frac{\partial }{\partial \mathrm{t}}}\left(\mathrm{v}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}{\displaystyle \frac{\partial \dot{\mathrm{v}}}{\partial \mathrm{v}}}\right)\\ & =\left(2{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}+\mathrm{v}{\displaystyle \frac{{\partial }^2\mathrm{i}}{\partial {\mathrm{v}}^2}}\right)\ddot{\mathrm{v}}+\left(3{\displaystyle \frac{{\partial }^2\mathrm{i}}{\partial {\mathrm{v}}^2}}+\mathrm{v}{\displaystyle \frac{{\partial }^3\mathrm{i}}{\partial {\mathrm{v}}^3}}\right){\dot{\mathrm{v}}}^2+2\dot{\mathrm{v}}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}{\displaystyle \frac{\partial \dot{\mathrm{v}}}{\partial \mathrm{v}}}+\mathrm{v}\dot{\mathrm{v}}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}{\displaystyle \frac{{\partial }^2\dot{\mathrm{v}}}{\partial {\mathrm{v}}^2}}+{\displaystyle \frac{\partial }{\partial \mathrm{t}}}\left(\mathrm{v}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}{\displaystyle \frac{\partial \dot{\mathrm{v}}}{\partial \mathrm{v}}}\right)\end{array}$$

(9)

or

$$\ddot{\mathrm{x}}={\displaystyle \frac{{\partial }^2\mathrm{p}}{\partial {\mathrm{v}}^2}}\ddot{\mathrm{v}}+{\displaystyle \frac{{\partial }^3\mathrm{p}}{\partial {\mathrm{v}}^3}}{\dot{\mathrm{v}}}^2+2\dot{\mathrm{v}}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}{\displaystyle \frac{\partial \dot{\mathrm{v}}}{\partial \mathrm{v}}}+\mathrm{v}\dot{\mathrm{v}}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}{\displaystyle \frac{{\partial }^2\dot{\mathrm{v}}}{\partial {\mathrm{v}}^2}}+{\displaystyle \frac{\partial }{\partial \mathrm{t}}}\left(\mathrm{v}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}{\displaystyle \frac{\partial \dot{\mathrm{v}}}{\partial \mathrm{v}}}\right)$$

(10)

in which

$$\begin{array}{l}{\displaystyle \frac{{\partial }^2\mathrm{p}}{\partial {\mathrm{v}}^2}}=2{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}+\mathrm{v}{\displaystyle \frac{{\partial }^2\mathrm{i}}{\partial {\mathrm{v}}^2}}\\ {\displaystyle \frac{{\partial }^3\mathrm{p}}{\partial {\mathrm{v}}^3}}=3{\displaystyle \frac{{\partial }^2\mathrm{i}}{\partial {\mathrm{v}}^2}}+\mathrm{v}{\displaystyle \frac{{\partial }^3\mathrm{i}}{\partial {\mathrm{v}}^3}}\end{array}$$

(11)

By definition of $\mathrm{u}=1-{\mathrm{T}}_{\mathrm{d}}$, we get

$$\ddot{\mathrm{v}}={\displaystyle \frac{1}{{\mathrm{C}}_{\mathrm{i}}}}\left(\dot{\mathrm{i}}-{\dot{\mathrm{i}}}_{\mathrm{l}\mathrm{i}}\right)={\displaystyle \frac{1}{{\mathrm{C}}_{\mathrm{i}}}}\left({\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}\dot{\mathrm{v}}-{\displaystyle \frac{\mathrm{v}}{{\mathrm{L}}_{\mathrm{i}}}}+{\displaystyle \frac{{\mathrm{V}}_{\mathrm{c}}}{{\mathrm{L}}_{\mathrm{i}}}}\mathrm{u}\right)$$

(12)

Therefore

$$\ddot{\mathrm{x}}={\displaystyle \frac{{\partial }^3\mathrm{p}}{\partial {\mathrm{v}}^3}}{\dot{\mathrm{v}}}^2+{\displaystyle \frac{1}{{\mathrm{C}}_{\mathrm{i}}}}{\displaystyle \frac{{\partial }^2\mathrm{p}}{\partial {\mathrm{v}}^2}}\left({\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}\dot{\mathrm{v}}-{\displaystyle \frac{\mathrm{v}}{{\mathrm{L}}_{\mathrm{i}}}}\right)+{\displaystyle \frac{{\mathrm{V}}_{\mathrm{c}}}{{\mathrm{C}}_{\mathrm{i}}{\mathrm{L}}_{\mathrm{i}}}}{\displaystyle \frac{{\partial }^2\mathrm{p}}{\partial {\mathrm{v}}^2}}\mathrm{u}+\mathrm{\Delta }$$

(13)

where $\mathrm{\Delta }$ is an unknown variable considered as uncertainty due to the unknown variable ${\displaystyle \frac{\partial \dot{\mathrm{v}}}{\partial \mathrm{v}}}$.

$$\mathrm{\Delta }=\mathrm{v}\dot{\mathrm{v}}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}{\displaystyle \frac{{\partial }^2\dot{\mathrm{v}}}{\partial {\mathrm{v}}^2}}+{\displaystyle \frac{\partial }{\partial \mathrm{t}}}\left(\mathrm{v}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}{\displaystyle \frac{\partial \dot{\mathrm{v}}}{\partial \mathrm{v}}}\right)+2\dot{\mathrm{v}}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}{\displaystyle \frac{\partial \dot{\mathrm{v}}}{\partial \mathrm{v}}}$$

(14)

Next, we calculate the variables ${\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}$, ${\displaystyle \frac{{\partial }^2\mathrm{i}}{\partial {\mathrm{v}}^2}}$, and ${\displaystyle \frac{{\partial }^3\mathrm{i}}{\partial {\mathrm{v}}^3}}$. First, we define the function $\mathrm{f}(\mathrm{i},\mathrm{v})$:

$$\mathrm{f}(\mathrm{i},\mathrm{v})={\mathrm{I}}_{\mathrm{p}\mathrm{h}\mathrm{g}}-{\mathrm{I}}_{\mathrm{r}\mathrm{g}}\exp \left({\displaystyle \frac{\mathrm{v}+{\mathrm{R}}_{\mathrm{s}\mathrm{g}}\mathrm{i}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}}\right)+{\mathrm{I}}_{\mathrm{r}\mathrm{g}}-{\displaystyle \frac{\mathrm{v}+{\mathrm{R}}_{\mathrm{s}\mathrm{g}}\mathrm{i}}{{\mathrm{R}}_{\mathrm{s}\mathrm{h}\mathrm{g}}}}-\mathrm{i}=0$$

(15)

Then, we have

$${\displaystyle \frac{\partial \mathrm{f}}{\partial \mathrm{v}}}+{\displaystyle \frac{\partial \mathrm{f}}{\partial \mathrm{i}}}{\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}=0$$

(16)

Thus, one can write

$${\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}=-{\displaystyle \frac{\frac{\partial \mathrm{f}}{\partial \mathrm{v}}}{\frac{\partial \mathrm{f}}{\partial \mathrm{i}}}}=-{\displaystyle \frac{\frac{{\mathrm{I}}_{\mathrm{r}\mathrm{g}}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\exp \left(\frac{\mathrm{v}+{\mathrm{R}}_{\mathrm{s}\mathrm{g}}\mathrm{i}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\right)+\frac{1}{{\mathrm{R}}_{\mathrm{s}\mathrm{h}\mathrm{g}}}}{\frac{{\mathrm{I}}_{\mathrm{r}\mathrm{g}}{\mathrm{R}}_{\mathrm{s}\mathrm{g}}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\exp \left(\frac{\mathrm{v}+{\mathrm{R}}_{\mathrm{s}\mathrm{g}}\mathrm{i}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\right)+\frac{{\mathrm{R}}_{\mathrm{s}\mathrm{g}}}{{\mathrm{R}}_{\mathrm{s}\mathrm{h}\mathrm{g}}}+1}}$$

(17)

Now, considering the equation:

$${\displaystyle \frac{{\partial }^2\mathrm{f}}{\partial {\mathrm{v}}^2}}+{\displaystyle \frac{\partial }{\partial \mathrm{v}}}\left({\displaystyle \frac{\partial \mathrm{f}}{\partial \mathrm{i}}}\right){\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}+{\displaystyle \frac{\partial \mathrm{f}}{\partial \mathrm{i}}}{\displaystyle \frac{{\partial }^2\mathrm{i}}{\partial {\mathrm{v}}^2}}=0$$

(18)

Therefore

$$\begin{array}{l}{\displaystyle \frac{{\partial }^2\mathrm{i}}{\partial {\mathrm{v}}^2}}=-{\displaystyle \frac{\frac{{\partial }^2\mathrm{f}}{\partial {\mathrm{v}}^2}+\frac{\partial }{\partial \mathrm{v}}\left(\frac{\partial \mathrm{f}}{\partial \mathrm{i}}\right)\frac{\partial \mathrm{i}}{\partial \mathrm{v}}}{\frac{\partial \mathrm{f}}{\partial \mathrm{i}}}}=-{\displaystyle \frac{\frac{{\mathrm{I}}_{\mathrm{r}\mathrm{g}}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}^2}\exp \left(\frac{\mathrm{v}+{\mathrm{R}}_{\mathrm{s}\mathrm{g}}\mathrm{i}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\right)+\frac{{\mathrm{I}}_{\mathrm{r}\mathrm{g}}{\mathrm{R}}_{\mathrm{s}\mathrm{g}}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}^2}\exp \left(\frac{\mathrm{v}+{\mathrm{R}}_{\mathrm{s}\mathrm{g}}\mathrm{i}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\right)\frac{\partial \mathrm{i}}{\partial \mathrm{v}}}{\left(\frac{{\mathrm{I}}_{\mathrm{r}\mathrm{g}}{\mathrm{R}}_{\mathrm{s}\mathrm{g}}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\exp \left(\frac{\mathrm{v}+{\mathrm{R}}_{\mathrm{s}\mathrm{g}}\mathrm{i}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\right)+\frac{{\mathrm{R}}_{\mathrm{s}\mathrm{g}}}{{\mathrm{R}}_{\mathrm{s}\mathrm{h}\mathrm{g}}}+1\right)}}\\ =-{\displaystyle \frac{\frac{{\mathrm{I}}_{\mathrm{r}\mathrm{g}}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}^2}\exp \left(\frac{\mathrm{v}+{\mathrm{R}}_{\mathrm{s}\mathrm{g}}\mathrm{i}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\right)\left(1+{\mathrm{R}}_{\mathrm{s}\mathrm{g}}\frac{\partial \mathrm{i}}{\partial \mathrm{v}}\right)}{\left(\frac{{\mathrm{I}}_{\mathrm{r}\mathrm{g}}{\mathrm{R}}_{\mathrm{s}\mathrm{g}}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\exp \left(\frac{\mathrm{v}+{\mathrm{R}}_{\mathrm{s}\mathrm{g}}\mathrm{i}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\right)+\frac{{\mathrm{R}}_{\mathrm{s}\mathrm{g}}}{{\mathrm{R}}_{\mathrm{s}\mathrm{h}\mathrm{g}}}+1\right)}}={\displaystyle \frac{-\frac{{\mathrm{I}}_{\mathrm{r}\mathrm{g}}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}^2}\exp \left(\frac{\mathrm{v}+{\mathrm{R}}_{\mathrm{s}\mathrm{g}}\mathrm{i}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\right)}{{\left(\frac{{\mathrm{I}}_{\mathrm{r}\mathrm{g}}{\mathrm{R}}_{\mathrm{s}\mathrm{g}}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\exp \left(\frac{\mathrm{v}+{\mathrm{R}}_{\mathrm{s}\mathrm{g}}\mathrm{i}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\right)+\frac{{\mathrm{R}}_{\mathrm{s}\mathrm{g}}}{{\mathrm{R}}_{\mathrm{s}\mathrm{h}\mathrm{g}}}+1\right)}^2}}\end{array}$$

(19)

Finally

$${\displaystyle \frac{{\partial }^3\mathrm{f}}{\partial {\mathrm{v}}^3}}+{\displaystyle \frac{\partial }{\partial \mathrm{v}}}\left({\displaystyle \frac{{\partial }^2\mathrm{f}}{\partial \mathrm{v}\partial \mathrm{i}}}\right){\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}+\left({\displaystyle \frac{{\partial }^2\mathrm{f}}{\partial \mathrm{v}\partial \mathrm{i}}}\right){\displaystyle \frac{{\partial }^2\mathrm{i}}{\partial {\mathrm{v}}^2}}+\left({\displaystyle \frac{{\partial }^2\mathrm{f}}{\partial \mathrm{v}\partial \mathrm{i}}}\right){\displaystyle \frac{{\partial }^2\mathrm{i}}{\partial {\mathrm{v}}^2}}+{\displaystyle \frac{\partial \mathrm{f}}{\partial \mathrm{i}}}{\displaystyle \frac{{\partial }^3\mathrm{i}}{\partial {\mathrm{v}}^3}}=0$$

(20)

Then

$$\begin{array}{l}{\displaystyle \frac{{\partial }^3\mathrm{i}}{\partial {\mathrm{v}}^3}}=-{\displaystyle \frac{\frac{{\partial }^3\mathrm{f}}{\partial {\mathrm{v}}^3}+\frac{\partial }{\partial \mathrm{v}}\left(\frac{{\partial }^2\mathrm{f}}{\partial \mathrm{v}\partial \mathrm{i}}\right)\frac{\partial \mathrm{i}}{\partial \mathrm{v}}+2\left(\frac{{\partial }^2\mathrm{f}}{\partial \mathrm{v}\partial \mathrm{i}}\right)\frac{{\partial }^2\mathrm{i}}{\partial {\mathrm{v}}^2}}{\frac{\partial \mathrm{f}}{\partial \mathrm{i}}}}\\ =-{\displaystyle \frac{\frac{{\mathrm{I}}_{\mathrm{r}\mathrm{g}}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}^3}\exp \left(\frac{\mathrm{v}+{\mathrm{R}}_{\mathrm{s}\mathrm{g}}\mathrm{i}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\right)+\frac{{\mathrm{I}}_{\mathrm{r}\mathrm{g}}{\mathrm{R}}_{\mathrm{s}\mathrm{g}}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}^3}\exp \left(\frac{\mathrm{v}+{\mathrm{R}}_{\mathrm{s}\mathrm{g}}\mathrm{i}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\right)\frac{\partial \mathrm{i}}{\partial \mathrm{v}}+2\frac{{\mathrm{I}}_{\mathrm{r}\mathrm{g}}{\mathrm{R}}_{\mathrm{s}\mathrm{g}}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}^2}\exp \left(\frac{\mathrm{v}+{\mathrm{R}}_{\mathrm{s}\mathrm{g}}\mathrm{i}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\right)\frac{{\partial }^2\mathrm{i}}{\partial {\mathrm{v}}^2}}{\left(\frac{{\mathrm{I}}_{\mathrm{r}\mathrm{g}}{\mathrm{R}}_{\mathrm{s}\mathrm{g}}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\exp \left(\frac{\mathrm{v}+{\mathrm{R}}_{\mathrm{s}\mathrm{g}}\mathrm{i}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\right)+\frac{{\mathrm{R}}_{\mathrm{s}\mathrm{g}}}{{\mathrm{R}}_{\mathrm{s}\mathrm{h}\mathrm{g}}}+1\right)}}\\ =-{\displaystyle \frac{\frac{{\mathrm{I}}_{\mathrm{r}\mathrm{g}}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}^3}\exp \left(\frac{\mathrm{v}+{\mathrm{R}}_{\mathrm{s}\mathrm{g}}\mathrm{i}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\right)\left(1+{\mathrm{R}}_{\mathrm{s}\mathrm{g}}\frac{\partial \mathrm{i}}{\partial \mathrm{v}}\right)+2\frac{{\mathrm{I}}_{\mathrm{r}\mathrm{g}}{\mathrm{R}}_{\mathrm{s}\mathrm{g}}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}^2}\exp \left(\frac{\mathrm{v}+{\mathrm{R}}_{\mathrm{s}\mathrm{g}}\mathrm{i}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\right)\frac{{\partial }^2\mathrm{i}}{\partial {\mathrm{v}}^2}}{\left(\frac{{\mathrm{I}}_{\mathrm{r}\mathrm{g}}{\mathrm{R}}_{\mathrm{s}\mathrm{g}}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\exp \left(\frac{\mathrm{v}+{\mathrm{R}}_{\mathrm{s}\mathrm{g}}\mathrm{i}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\right)+\frac{{\mathrm{R}}_{\mathrm{s}\mathrm{g}}}{{\mathrm{R}}_{\mathrm{s}\mathrm{h}\mathrm{g}}}+1\right)}}\\ ={\displaystyle \frac{\frac{{\mathrm{I}}_{\mathrm{r}\mathrm{g}}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}^3}\exp \left(\frac{\mathrm{v}+{\mathrm{R}}_{\mathrm{s}\mathrm{g}}\mathrm{i}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\right)\left(\frac{{\mathrm{I}}_{\mathrm{r}\mathrm{g}}{\mathrm{R}}_{\mathrm{s}\mathrm{g}}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\exp \left(\frac{\mathrm{v}+{\mathrm{R}}_{\mathrm{s}\mathrm{g}}\mathrm{i}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\right)-\frac{{\mathrm{R}}_{\mathrm{s}\mathrm{g}}}{{\mathrm{R}}_{\mathrm{s}\mathrm{h}\mathrm{g}}}-1\right)}{{\left(\frac{{\mathrm{I}}_{\mathrm{r}\mathrm{g}}{\mathrm{R}}_{\mathrm{s}\mathrm{g}}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\exp \left(\frac{\mathrm{v}+{\mathrm{R}}_{\mathrm{s}\mathrm{g}}\mathrm{i}}{{\mathrm{V}}_{\mathrm{T}\mathrm{g}}}\right)+\frac{{\mathrm{R}}_{\mathrm{s}\mathrm{g}}}{{\mathrm{R}}_{\mathrm{s}\mathrm{h}\mathrm{g}}}+1\right)}^3}}\end{array}$$

(21)

Now, we applied the following state feedback to Equation (13), with a new input variable ${\mathrm{v}}_{\mathrm{n}}$.

$$\mathrm{u}=-{\left({\displaystyle \frac{{\mathrm{V}}_{\mathrm{c}}}{{\mathrm{C}}_{\mathrm{i}}{\mathrm{L}}_{\mathrm{i}}}}{\displaystyle \frac{{\partial }^2\mathrm{p}}{\partial {\mathrm{v}}^2}}\right)}^{-1}\left({\displaystyle \frac{{\partial }^3\mathrm{p}}{\partial {\mathrm{v}}^3}}{\dot{\mathrm{v}}}^2+{\displaystyle \frac{1}{{\mathrm{C}}_{\mathrm{i}}}}{\displaystyle \frac{{\partial }^2\mathrm{p}}{\partial {\mathrm{v}}^2}}\left({\displaystyle \frac{\partial \mathrm{i}}{\partial \mathrm{v}}}\dot{\mathrm{v}}-{\displaystyle \frac{\mathrm{v}}{{\mathrm{L}}_{\mathrm{i}}}}\right)+{\mathrm{a}}_1\mathrm{x}+{\mathrm{a}}_2\dot{\mathrm{x}}-{\mathrm{v}}_{\mathrm{n}}\right)$$

(22)

We have

$$\ddot{\mathrm{x}}=-{\mathrm{a}}_1\mathrm{x}-{\mathrm{a}}_2\dot{\mathrm{x}}+{\mathrm{v}}_{\mathrm{n}}+\mathrm{\Delta }$$

(23)

Now, we define the state variables ${\mathrm{x}}_1=\mathrm{x},{\mathrm{x}}_2=\dot{\mathrm{x}}$, and therefore

$$\begin{array}{l}\dot{\mathrm{x}}=\mathrm{A}\mathrm{x}+\mathrm{B}{\mathrm{v}}_{\mathrm{n}}+\mathrm{B}\mathrm{\Delta }\\ \mathrm{x}=\left[\begin{array}{c}{\mathrm{x}}_1\\ {\mathrm{x}}_2\end{array}\right],\mathrm{A}=\left[\begin{array}{cc}0& 1\\ -{\mathrm{a}}_1& -{\mathrm{a}}_2\end{array}\right],\mathrm{B}=\left[\begin{array}{c}0\\ 1\end{array}\right]\end{array}$$

(24)

4. Observer and Controller Design

The parameters ${\mathrm{a}}_1$ and ${\mathrm{a}}_2$ are chosen such that matrix $\mathrm{A}$ is Hurwitz and pair $(\mathrm{A},\mathrm{B})$ is controllable. Then, for any matrix $\mathrm{Q}\in {\mathrm{R}}^{2\times 2}$, there exists matrix $\mathrm{P}\in {\mathrm{R}}^{2\times 2}$ that satisfies the following Lyapunov equation [20]:

$$\mathrm{Q}=-{\mathrm{A}}^{\mathrm{T}}\mathrm{P}-\mathrm{P}\mathrm{A}$$

(25)

Such that both matrixes $\mathrm{Q}\in {\mathrm{R}}^{2\times 2}$ and $\mathrm{P}\in {\mathrm{R}}^{2\times 2}$ are symmetric and SPD.

4.1. Proposed SMO

To achieve MPPT, the state variables ${\mathrm{x}}_1$ and ${\mathrm{x}}_2$ in Equation (24) should be zero. To implement DSMC, the unavailable variable ${\dot{\mathrm{x}}}_2$ should be estimated. To this end, the following SMO with error estimation $\mathrm{e}=\mathrm{x}-\widehat{\mathrm{x}}$ and auxiliary signal $\mathrm{z}$ is proposed.

$$\left\{\begin{array}{l}\dot{\mathrm{z}}(\mathrm{t})=\mathrm{e}-\mathrm{z}(\mathrm{t})\\ \dot{\widehat{\mathrm{x}}}(\mathrm{t})=\mathrm{A}\widehat{\mathrm{x}}+\mathrm{B}{\mathrm{v}}_{\mathrm{n}}+{\mathrm{P}}^{-1}\left(\mathrm{P}\mathrm{B}\mathrm{D}+\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\mathrm{e})\right)+\mathrm{z}(\mathrm{t})\\ \mathrm{z}(\mathrm{t})={[{\mathrm{z}}_1,{\mathrm{z}}_2]}^{\mathrm{T}},\widehat{\mathrm{x}}(\mathrm{t})={[{\widehat{\mathrm{x}}}_1,{\widehat{\mathrm{x}}}_2]}^{\mathrm{T}},\left|\mathrm{\Delta }\right|\le \mathrm{D}\end{array}\right.$$

(26)

Remark 1.

Considering a bound $\mathrm{D}$ for uncertain $\mathrm{\Delta }$ has no problem. It is very popular considering such a bound, because after proofing the closed-loop stability, all the variables of the systems are bounded (see Section 7 of [20]).

Theorem 1.

The error estimation converges to zero in the above proposed SMO.

Proof.

The derivative of the error estimation can be obtained as follows using Equations (24) and (26):

$$\dot{\mathrm{e}}(\mathrm{t})=\mathrm{A}\mathrm{e}+\mathrm{B}\mathrm{\Delta }-\mathrm{z}(\mathrm{t})+-{\mathrm{P}}^{-1}\left(\mathrm{P}\mathrm{B}\mathrm{D}+\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\mathrm{e})\right)$$

(27)

Using the Lyapunov function as

$$\mathrm{V}(\mathrm{t})={\displaystyle \frac{1}{2}}\mathrm{z}{(\mathrm{t})}^{\mathrm{T}}\mathrm{P}\mathrm{z}(\mathrm{t})+{\displaystyle \frac{1}{2}}\mathrm{e}{(\mathrm{t})}^{\mathrm{T}}\mathrm{P}\mathrm{e}(\mathrm{t})$$

(28)

Then, due to the symmetric of matrix $\mathrm{P}$, we have

$$\dot{\mathrm{V}}={\displaystyle \frac{1}{2}}{\dot{\mathrm{z}}}^{\mathrm{T}}\mathrm{P}\mathrm{z}+{\displaystyle \frac{1}{2}}{\mathrm{z}}^{\mathrm{T}}\mathrm{P}\dot{\mathrm{z}}+{\displaystyle \frac{1}{2}}{\dot{\mathrm{e}}}^{\mathrm{T}}\mathrm{P}\mathrm{e}+{\displaystyle \frac{1}{2}}{\mathrm{e}}^{\mathrm{T}}\mathrm{P}\dot{\mathrm{e}}={\dot{\mathrm{z}}}^{\mathrm{T}}\mathrm{P}\mathrm{z}+{\dot{\mathrm{e}}}^{\mathrm{T}}\mathrm{P}\mathrm{e}$$

(29)

Thus, one can obtain

$$\dot{\mathrm{V}}={\left(\mathrm{e}-\mathrm{z}\right)}^{\mathrm{T}}\mathrm{P}\mathrm{z}+{\left(\mathrm{A}\mathrm{e}+\mathrm{B}\mathrm{\Delta }-\mathrm{z}-\mathrm{B}\mathrm{D}-{\mathrm{P}}^{-1}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\mathrm{e})\right)}^{\mathrm{T}}\mathrm{P}\mathrm{e}$$

(30)

or

$$\begin{array}{ll}\dot{\mathrm{V}}& ={\mathrm{e}}^{\mathrm{T}}\mathrm{P}\mathrm{z}-{\mathrm{z}}^{\mathrm{T}}\mathrm{P}\mathrm{z}+{\mathrm{e}}^{\mathrm{T}}{\mathrm{A}}^{\mathrm{T}}\mathrm{P}\mathrm{e}+{\mathrm{B}}^{\mathrm{T}}\mathrm{P}\mathrm{e}\mathrm{\Delta }-{\mathrm{z}}^{\mathrm{T}}\mathrm{P}\mathrm{e}-{\mathrm{B}}^{\mathrm{T}}\mathrm{P}\mathrm{e}\mathrm{D}-\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}{(\mathrm{e})}^{\mathrm{T}}{\mathrm{P}}^{-1}\mathrm{P}\mathrm{e}\\ & ={\mathrm{e}}^{\mathrm{T}}\mathrm{P}\mathrm{z}-{\mathrm{z}}^{\mathrm{T}}\mathrm{P}\mathrm{z}+{\mathrm{e}}^{\mathrm{T}}{\mathrm{A}}^{\mathrm{T}}\mathrm{P}\mathrm{e}+{\mathrm{B}}^{\mathrm{T}}\mathrm{P}\mathrm{e}\mathrm{\Delta }-{\mathrm{e}}^{\mathrm{T}}\mathrm{P}\mathrm{z}-{\mathrm{e}}^{\mathrm{T}}\mathrm{P}\mathrm{B}\mathrm{D}-{\mathrm{e}}^{\mathrm{T}}{\mathrm{P}}^{-1}\mathrm{P}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\mathrm{e})\\ & =-{\mathrm{z}}^{\mathrm{T}}\mathrm{P}\mathrm{z}+{\mathrm{e}}^{\mathrm{T}}{\mathrm{A}}^{\mathrm{T}}\mathrm{P}\mathrm{e}+{\mathrm{e}}^{\mathrm{T}}\mathrm{P}\mathrm{B}(\mathrm{\Delta }-\mathrm{D})-{\mathrm{e}}^{\mathrm{T}}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\mathrm{e})\\ & =-{\mathrm{z}}^{\mathrm{T}}\mathrm{P}\mathrm{z}+{\displaystyle \frac{1}{2}}{\mathrm{e}}^{\mathrm{T}}{\mathrm{A}}^{\mathrm{T}}\mathrm{P}\mathrm{e}+{\displaystyle \frac{1}{2}}{\mathrm{e}}^{\mathrm{T}}\mathrm{P}\mathrm{A}\mathrm{e}+{\mathrm{e}}^{\mathrm{T}}\mathrm{P}\mathrm{B}(\mathrm{\Delta }-\mathrm{D})-{\mathrm{e}}^{\mathrm{T}}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\mathrm{e})\end{array}$$

(31)

Leading to

$$\dot{\mathrm{V}}=-{\mathrm{z}}^{\mathrm{T}}\mathrm{P}\mathrm{z}+{\displaystyle \frac{1}{2}}{\mathrm{e}}^{\mathrm{T}}\mathrm{Q}\mathrm{e}+{\mathrm{e}}^{\mathrm{T}}\mathrm{P}\mathrm{B}\left(\mathrm{\Delta }-\mathrm{D}\right)-{\mathrm{e}}^{\mathrm{T}}\mathrm{s}\mathrm{i}\mathrm{e}\mathrm{n}(\mathrm{e})$$

(32)

Using ${\mathrm{\lambda }}_{\min }$ as the minimum eigenvalue and referring to the following inequalities

$$\begin{array}{l}-{\mathrm{e}}^{\mathrm{T}}\mathrm{Q}\mathrm{e}\le -{\mathrm{\lambda }}_{\min }(\mathrm{Q}) {‖\mathrm{e}‖}^2\le 0\\ -{\mathrm{z}}^{\mathrm{T}}\mathrm{P}\mathrm{z}\le -{\mathrm{\lambda }}_{\min }(\mathrm{P}) {‖\mathrm{z}‖}^2\le 0\end{array}$$

(33)

Consequently

$$\begin{array}{ll}\dot{\mathrm{V}}& \le -{\mathrm{\lambda }}_{\min }(\mathrm{P}) {‖\mathrm{z}‖}^2-{\displaystyle \frac{1}{2}}{\mathrm{\lambda }}_{\min }(\mathrm{Q}) {‖\mathrm{e}‖}^2+\left|{\mathrm{e}}^{\mathrm{T}}\mathrm{P}\mathrm{B}\right| \left( \left|\mathrm{\Delta }\right|-\mathrm{D} \right)-{\displaystyle \sum _{\mathrm{i}=1}^2\left|{\mathrm{e}}_{\mathrm{i}}\right|}\\ & \le -{\mathrm{\lambda }}_{\min }(\mathrm{P}) {‖\mathrm{z}‖}^2-{\displaystyle \frac{1}{2}}{\mathrm{\lambda }}_{\min }(\mathrm{Q}) {‖\mathrm{e}‖}^2\end{array}$$

(34)

Then, the error dynamic $\mathrm{e}$ and axillary signal $\mathrm{z}$ converge to zero. □

4.2. Proposed DSMC

To address uncertainty and achieve a smooth duty cycle, DSMC is proposed and designed. Moreover, CSMC is also designed for comparison with DSMC using the same SMO in both approaches in order to carry out a fair comparison. According to the observer Equation (26), the following equations can be concluded:

$$\begin{array}{l}{\dot{\widehat{\mathrm{x}}}}_1={\mathrm{z}}_1+{\widehat{\mathrm{x}}}_2+{\mathrm{C}}_1{\mathrm{P}}^{-1}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\mathrm{e}),{\mathrm{C}}_1=[1,0]\\ {\dot{\widehat{\mathrm{x}}}}_2={\mathrm{z}}_2-{\mathrm{a}}_1{\widehat{\mathrm{x}}}_1-{\mathrm{a}}_2{\widehat{\mathrm{x}}}_2+{\mathrm{v}}_{\mathrm{n}}+\mathrm{D}+{\mathrm{C}}_2{\mathrm{P}}^{-1}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\mathrm{e}),{\mathrm{C}}_2=[0,1]\\ {\ddot{\widehat{\mathrm{x}}}}_2={\dot{\mathrm{z}}}_2-{\mathrm{a}}_1{\dot{\widehat{\mathrm{x}}}}_1-{\mathrm{a}}_2{\dot{\widehat{\mathrm{x}}}}_2+{\dot{\mathrm{v}}}_{\mathrm{n}}=-{\mathrm{z}}_2+{\mathrm{C}}_2\mathrm{e}-{\mathrm{a}}_1{\dot{\widehat{\mathrm{x}}}}_1-{\mathrm{a}}_2{\dot{\widehat{\mathrm{x}}}}_2+{\dot{\mathrm{v}}}_{\mathrm{n}}\end{array}$$

(35)

Then, in DSMC, the sliding variable is defined as

$$\mathrm{s}={\mathrm{\lambda }}_1{\widehat{\mathrm{x}}}_1+{\mathrm{\lambda }}_2{\widehat{\mathrm{x}}}_2+{\mathrm{\lambda }}_3{\dot{\widehat{\mathrm{x}}}}_2$$

(36)

Theorem 2.

The equation of sliding surface (36) reaches zero in finite time using the following input.

$${\dot{\mathrm{v}}}_{\mathrm{n}}=-{\displaystyle \frac{{\mathrm{\lambda }}_1{\dot{\widehat{\mathrm{x}}}}_1+{\mathrm{\lambda }}_2{\dot{\widehat{\mathrm{x}}}}_2+{\mathrm{\lambda }}_3\left(-{\mathrm{z}}_2+{\mathrm{C}}_2\mathrm{e}-{\mathrm{a}}_1{\dot{\widehat{\mathrm{x}}}}_1-{\mathrm{a}}_2{\dot{\widehat{\mathrm{x}}}}_2\right)+{\mathrm{\eta }}_1\mathrm{s}+{\mathrm{\eta }}_2\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\mathrm{s})}{{\mathrm{\lambda }}_3}}$$

(37)

Proof.

Equation (36) leads to

$$\dot{\mathrm{s}}={\mathrm{\lambda }}_1{\dot{\widehat{\mathrm{x}}}}_1+{\mathrm{\lambda }}_2{\dot{\widehat{\mathrm{x}}}}_2+{\mathrm{\lambda }}_3{\ddot{\widehat{\mathrm{x}}}}_2={\mathrm{\lambda }}_1{\dot{\widehat{\mathrm{x}}}}_1+{\mathrm{\lambda }}_2{\dot{\widehat{\mathrm{x}}}}_2+{\mathrm{\lambda }}_3(-{\mathrm{z}}_2+{\mathrm{C}}_2\mathrm{e}-{\mathrm{a}}_1{\dot{\widehat{\mathrm{x}}}}_1-{\mathrm{a}}_2{\dot{\widehat{\mathrm{x}}}}_2+{\dot{\mathrm{v}}}_{\mathrm{n}})$$

(38)

Using Equations (37) and (38) results in

$$\dot{\mathrm{s}}=-{\mathrm{\eta }}_2 \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\mathrm{s})-{\mathrm{\eta }}_1 \mathrm{s}$$

(39)

Suppose $\mathrm{V}=0.5{\mathrm{s}}^2$ as the Lyapunov function

$$\dot{\mathrm{V}}=\mathrm{s}\dot{\mathrm{s}}=\mathrm{s}\left(-{\mathrm{\eta }}_2 \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\mathrm{s})-{\mathrm{\eta }}_1 \mathrm{s}\right)=-{\mathrm{\eta }}_2\left|\mathrm{s}\right|-{\mathrm{\eta }}_1{\mathrm{s}}^2\le -{\mathrm{\eta }}_2\left|\mathrm{s}\right|$$

(40)

Then ${\mathrm{t}}_{\mathrm{f}}\le {\displaystyle \frac{\left|\mathrm{s}(0)\right|}{{\mathrm{\eta }}_2}}$ is the finite reaching time to the sliding surface [21]. □

4.3. CSMC Design

In CSMC, the sliding variable is defined as

$$\mathrm{s}={\mathrm{\lambda }}_1{\widehat{\mathrm{x}}}_1+{\mathrm{\lambda }}_2{\widehat{\mathrm{x}}}_2$$

(41)

Theorem 3.

The equation of sliding surface (41) reaches zero in finite time using the following input.

$${\mathrm{v}}_{\mathrm{n}}=-{\displaystyle \frac{{\mathrm{\lambda }}_1{\dot{\widehat{\mathrm{x}}}}_1+{\mathrm{\lambda }}_2\left({\mathrm{z}}_2-{\mathrm{a}}_1{\widehat{\mathrm{x}}}_1-{\mathrm{a}}_2{\widehat{\mathrm{x}}}_2+\mathrm{D}+{\mathrm{C}}_2{\mathrm{P}}^{-1}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\mathrm{e})\right)+{\mathrm{\eta }}_1\mathrm{s}+{\mathrm{\eta }}_2\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\mathrm{s})}{{\mathrm{\lambda }}_2}}$$

(42)

Proof.

Equation (41) leads to

$$\dot{\mathrm{s}}={\mathrm{\lambda }}_1{\dot{\widehat{\mathrm{x}}}}_1+{\mathrm{\lambda }}_2{\dot{\widehat{\mathrm{x}}}}_2={\mathrm{\lambda }}_1{\dot{\widehat{\mathrm{x}}}}_1+{\mathrm{\lambda }}_2\left({\mathrm{z}}_2-{\mathrm{a}}_1{\widehat{\mathrm{x}}}_1-{\mathrm{a}}_2{\widehat{\mathrm{x}}}_2+{\mathrm{v}}_{\mathrm{n}}+\mathrm{D}+{\mathrm{C}}_2{\mathrm{P}}^{-1}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\mathrm{e})\right)$$

(43)

Using Equations (42) and (43) results in

$$\dot{\mathrm{s}}=-{\mathrm{\eta }}_2 \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\mathrm{s})-{\mathrm{\eta }}_1 \mathrm{s}$$

(44)

Suppose $\mathrm{V}=0.5{\mathrm{s}}^2$ as the Lyapunov function.

$$\dot{\mathrm{V}}=\mathrm{s}\dot{\mathrm{s}}=\mathrm{s}\left(-{\mathrm{\eta }}_2 \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\mathrm{s})-{\mathrm{\eta }}_1 \mathrm{s}\right)=-{\mathrm{\eta }}_2\left|\mathrm{s}\right|-{\mathrm{\eta }}_1{\mathrm{s}}^2\le -{\mathrm{\eta }}_2\left|\mathrm{s}\right|$$

(45)

Then ${\mathrm{t}}_{\mathrm{f}}\le {\displaystyle \frac{\left|\mathrm{s}(0)\right|}{{\mathrm{\eta }}_2}}$ is the finite reaching time to the sliding surface [21]. □

5. Presentation of Simulation Results

The single solar panel PV cells parameters are listed in

Table 2. Single PV cell parameters.

Parameter	Value	Unit
${\mathrm{I}}_{\mathrm{r}}$	4.842	$\mathrm{\mu }\mathrm{A}$
${\mathrm{I}}_{\mathrm{s}\mathrm{c}}$	3.45	$\mathrm{A}$
${\mathrm{R}}_{\mathrm{s}}$	0.1124	$\mathrm{\Omega }$
${\mathrm{R}}_{\mathrm{s}\mathrm{h}}$	6500	$\mathrm{\Omega }$
${\mathrm{T}}_{\mathrm{r}}$	298.15	$°\mathrm{K}$

, which are extracted from TE500CR [52].

The IBC parameters are also denoted in

Table 3. The converter parameters.

Parameter	Value	Unit
${\mathrm{V}}_{\mathrm{c}}$	400	$\mathrm{V}$
${\mathrm{I}}_{\mathrm{c}}$	12.5	$\mathrm{A}$
${\mathrm{L}}_{\mathrm{i}}$	3.5	$\mathrm{m}\mathrm{H}$
${\mathrm{C}}_{\mathrm{i}}$	4700	$\mathrm{\mu }\mathrm{F}$
${\mathrm{C}}_{\mathrm{o}}$	470	$\mathrm{\mu }\mathrm{F}$

[4].

Additionally, we have chosen ${\mathrm{N}}_{\mathrm{p}}=5$, ${\mathrm{N}}_{\mathrm{s}}=5$, also $\mathrm{z}(0)={[0,0]}^{\mathrm{T}}$ and $\widehat{\mathrm{x}}(0)={[0,0]}^{\mathrm{T}}$, i.e., all the initial values of the observer are set to zero. Moreover, we have selected $\mathrm{i}(0)=1$, $\mathrm{v}(0)=2$, and ${\mathrm{i}}_{\mathrm{l}\mathrm{i}}(0)=5$ and chose ${\mathrm{a}}_1=5$ and ${\mathrm{a}}_2=10$. For a reliable comparison, two simulations are conducted using DSMC and CSMC, both based on the same SMO.

To show the performance of the approaches and according to Figure 4 and Figure 5, the solar irradiation and the temperature increase from $\mathrm{\lambda }=800 \mathrm{W}/{\mathrm{m}}^2$ to $\mathrm{\lambda }=1000 \mathrm{W}/{\mathrm{m}}^2$ and from $\mathrm{T}=298.15 °\mathrm{K}$ to $\mathrm{T}=325 °\mathrm{K}$ at times $10 \mathrm{s}$ and $20 \mathrm{s}$, respectively.

5.1. Example 1: The DSMC Proposed Method

The block diagram of the proposed DSMC is shown in Figure 6. Therefore, after the definition of state variables, a state feedback with a new input variable is implemented. After that, to estimate the added extra state, an SMO is used. Then, the derivative of this new input control is calculated. Finally, the integrator will remove the chattering, which is produced by the Signum function. Note that the sampling time is $0.0001 \mathrm{s}$ and the PWM frequency is 10,000$\mathrm{H}\mathrm{z}$.

The parameters ${\mathrm{\lambda }}_1=4$, ${\mathrm{\lambda }}_2=0.4$, ${\mathrm{\lambda }}_3=0.01$ are selected as sliding surface coefficients, and we chose ${\mathrm{\eta }}_1=20$ and ${\mathrm{\eta }}_2=2$. The initial value is selected as ${\mathrm{v}}_{\mathrm{n}}(0)=0$ to calculate the input control from Equation (37). The results are presented in Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11, all of them without chattering. Figure 7 shows the output power derivative with respect to voltage and its estimation, which are zero. It deviates from zero only when the solar irradiation and temperature are varying. Figure 8 shows the output current, which increases with rising solar irradiation and temperature, consistent with Equations (4) and (5). Figure 9 shows the output voltage, and Figure 10 presents the smooth duty cycle of the IBC (the output of PWM). The duty cycle also decreases to maintain the converter output voltage fixed at ${\mathrm{V}}_{\mathrm{c}}=400$, when the output voltage of PVGS increases. Figure 11 shows the sliding surface and its finite-time convergence to zero.

5.2. Example 2: The CSMC Approach

Here CSMC is used with the same parameters as the DSMC example. Hence, the coefficients of sliding surface (39) are set as ${\mathrm{\lambda }}_1=0.4$, ${\mathrm{\lambda }}_2=0.01$, and also ${\mathrm{\eta }}_1=20$ and ${\mathrm{\eta }}_2=2$. The results are shown in Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16.

Figure 12 shows the output power derivative with respect to voltage and its estimation, which are zero, deviating only when solar irradiation and temperature are varying. Figure 13 shows the output current, which also increases with rising solar irradiation and temperature, and is consistent with Equations (4) and (5). This is rational according to Equations (4) and (5). Figure 14 shows the output voltage, and Figure 15 also presents the duty cycle of the converter (the output of PWM). The duty cycle also decreases to maintain the converter output voltage fixed at ${\mathrm{V}}_{\mathrm{c}}=400$, when the output voltage of PVGS increases. Figure 16 shows the sliding surface and its finite-time convergence to zero. The noticeable chattering can be seen in Figure 15 and Figure 16.

5.3. Comparison Results: The DSMC and CSMC Approach

Figure 7 is similar to Figure 12; however, Figure 7 shows better efficiency in MPPT. Since the output power derivative with respect to voltage is completely zero in Figure 7, but it deviates from zero in Figure 12. The low amplitude in the observer in these figures is due to the Signum function in SMO. Furthermore, the duty cycle in Figure 10 is smooth, while in Figure 15, it exhibits high-frequency chattering, making it unsuitable for the IBC and cannot be applied to it.

Figure 17 shows the output powers. As one can see, the output power of the two approaches is same in amplitude, except that in DSMC, chattering is removed, but in CSMC, it is still available.

Moreover,

Table 4. Some statistical parameters.

Methods	DSMC			CSMC
Parameter	Min	Max	RMS	Min	Max	RMS
${\mathrm{x}}_1$	−2.4387	9.6121	0.5074	−1.4473	9.649	0.4012
${\widehat{\mathrm{x}}}_1$	−2.4537	3.8309	0.3238	−1.4577	1.81	0.1637
$\mathrm{i}$	1	12.0259	7.9905	1	12.0286	7.9976
$\mathrm{v}$	1.1489	325.3517	293.5982	1.1489	324.4607	293.3747
${\mathrm{T}}_{\mathrm{d}}$	0.1866	0.994	0.2737	0.1881	0.994	0.2736
$\mathrm{s}$	−0.0011	0.051	0.0012	−0.3021	0.2829	0.0961

compares some statistical parameters of DSMC and CMC, such as minimum (Min), maximum (Max), and RMS of all variables. As we can see, there are not many differences in most of these parameters, especially in the important variable duty cycle of ${\mathrm{T}}_{\mathrm{d}}$. Anyway, the parameters of the sliding surface in DSMC are much smaller than CSMC, i.e., the invariance property of DSMC is better.

Comparing these figures highlights the superior performance of the proposed DSMC and SMO. From a controller design perspective, DSMC outperforms CSMC in chattering suppression. Additionally, the simple design of DSMC compared to CSMC is evident.

6. Conclusions

This paper introduces a new method to regulate the smooth duty cycle of an increasing boost converter (IBC), which is positioned after a photovoltaic generator system (PVGS). By appropriately selecting the converter duty cycle, the maximum power point tracking (MPPT) of the PVGS can be achieved. Initially, a state feedback is used to linearize the system. However, the combined model of PVGS and IBC has some uncertainties. To address these uncertainties and ensure a smooth duty cycle, dynamic sliding mode control (DSMC), as a new robust chattering-free controller, is proposed. Also, a new structure for the sliding mode observer (SMO) is developed to implement the DSMC. The performance of this approach is compared with conventional sliding mode control (CSMC) using similar parameters with the same SMO. The comparison reveals that CSMC exhibits high-frequency chattering, while DSMC eliminates it. Additionally, DSMC is noted for its simplicity in concept and implementation. The closed-loop stability is ensured using Lyapunov theory.