In this work, a boost controller is used as the converter; its structure is illustrated in Figure 1 . Power converters’ dynamic models are most often nonlinear.

With such a converter, the goal is to establish an average voltage, $V_{bus\_DC\_ref}$, at the converter output that meets the following criteria: $V_{bus\_DC\_ref}\succ V_{PV}$ and $V_{bus\_DC\_ref}$ is adjustable.

It is assumed that the voltage from the PV is the voltage $V_{mp}$ at the maximum power point, obtained by the “perturb and observe” control [37] .

For the duty cycle $\mu =1$ switch $ST$ is closed and the diode $D$ is open. During this phase, the inductor $L_{PV}$ starts to accumulate energy. Indeed, the inductor is subjected to a constant voltage source $V_{PV}$. The transfer diode $D$ is blocked and the capacitor $C_{dc}$, initially charged, slowly discharges into the load $I_{load}$; the equivalent circuit is illustrated in Figure 2 .

The dynamic model for this phase is written as follows:

$\{\begin{array}{l}\frac{\text{d}{I}_{L_{pv}}}{\text{d}t}=\frac{-R_{pv}}{L_{pv}}{I}_{L_{pv}}+\frac{V_{pv}}{L_{pv}}\\ \frac{\text{d}{V}_{bus\_dc}}{\text{d}t}=-\frac{1}{C_{dc}}{I}_{load}\end{array}$ (1)

For the duty cycle $\mu =0$ the switch $ST$ is open and the diode $D$ is closed. Diode D starts to conduct to ensure the continuity of the current in the inductor $L_{PV}$. The energy accumulated by the inductor is then transferred to the load and to the capacitor which renews its voltage. The equivalent circuit is shown in Figure 3 .

The dynamic model for this phase is written as follows:

$\{\begin{array}{l}\frac{\text{d}{I}_{L_{pv}}}{\text{d}t}=\frac{-R_{pv}}{L_{pv}}{I}_{L_{pv}}+\frac{V_{pv}}{L_{pv}}-\frac{V_{bus\_DC}}{L_{pv}}\\ \frac{\text{d}{V}_{bus\_DC}}{\text{d}t}=-\frac{1}{C_{dc}}{I}_{load}+\frac{I_{L_{pv}}}{C_{dc}}\end{array}$(2)

When the switching frequency is high, it is possible to model the average dynamic behavior of these converters [38] [39] . Combining the subsystems of Equations (1) and (2), we obtain the average dynamic model of the system:

$\{\begin{array}{l}\frac{\text{d}{I}_{L_{pv}}}{\text{d}t}=\frac{-R_{pv}}{L_{pv}}{I}_{L_{pv}}+\frac{V_{pv}}{L_{pv}}-\frac{\left(1-\mu \right)}{L_{pv}}{V}_{bus\_DC}\\ \frac{\text{d}{V}_{bus\_DC}}{\text{d}t}=\frac{\left(1-\mu \right)}{C_{dc}}{I}_{L_{pv}}-\frac{1}{C_{dc}}{I}_{load}\end{array}$(3)

With $\mu \in \left[0,1\right]$.

By putting $x_1=I_{L_{pv}}$ et $x_2=V_{bus\_DC}$, Equation (3) takes the following form:

$\stackrel{\dot{}}{x}\left(t\right)=Ax\left(t\right)+g\left(x\right)\cdot \mu +E_d\cdot d$ (4)

With $A=\left(\begin{array}{cc}-R_{pv}/L_{pv}& -1/L_{pv}\\ 1/C_{dc}& 0\end{array}\right)$, $g\left(x\right)=\left(\begin{array}{c}{x}_2\left(t\right)/L_{pv}\\ -x_1\left(t\right)/C_{pv}\end{array}\right)$, $E_d=I_{2x2}$, $d=\left(\begin{array}{c}{V}_{pv}/L_{pv}\\ -I_{load}/C_{dc}\end{array}\right)$.

Equation (4) represents the nonlinear dynamic model of the system under investigation.

Let’s put $\zeta \left(t\right)=\left(\begin{array}{c}{\zeta }_1\left(t\right)\\ {\zeta }_2\left(t\right)\end{array}\right)=\left(\begin{array}{c}{x}_1\left(t\right)-x_{1\_ref}\left(t\right)\\ x_2\left(t\right)-x_{2\_ref}\left(t\right)\end{array}\right)$ the error between the nominal states and the set references.

The dynamic of the error becomes:

$\stackrel{\dot{}}{\zeta }\left(t\right)=\left(\begin{array}{c}{\stackrel{\dot{}}{x}}_1\left(t\right)\\ {\stackrel{\dot{}}{x}}_2\left(t\right)\end{array}\right)=\left(\begin{array}{c}\frac{-R_{pv}}{L_{pv}}\left({\zeta }_1\left(t\right)+x_{1\_ref}\left(t\right)\right)-\frac{\left(1-\mu \right)}{L_{pv}}\left({\zeta }_2\left(t\right)+x_{2\_ref}\left(t\right)\right)+\frac{V_{pv}}{L_{pv}}\\ \frac{\left(1-\mu \right)}{C_{dc}}\left({\zeta }_1\left(t\right)+x_{1\_ref}\left(t\right)\right)-\frac{1}{C_{dc}}{I}_{load}\end{array}\right)$ (11)

In steady state $\stackrel{\dot{}}{\zeta }\left(t\right)=\zeta \left(t\right)=0_{2x1}$, thus:

$\left(\begin{array}{c}\frac{-R_{pv}}{L_{pv}}{x}_{1\_ref}\left(t\right)-\frac{\left(1-\mu \right)}{L_{pv}}{x}_{2\_ref}\left(t\right)+\frac{V_{pv}}{L_{pv}}\\ \frac{\left(1-\mu \right)}{C_{dc}}{x}_{1\_ref}\left(t\right)-\frac{1}{C_{dc}}{I}_{load}\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right)$ (12)

At high frequencies $\mu$ is a constant. Thus, the PV voltage and the load current are expressed as:

$\left(\begin{array}{c}{V}_{pv}\\ I_{load}\end{array}\right)=\left(\begin{array}{c}{R}_{pv}{x}_{1\_ref}\left(t\right)+\left(1-\mu \right)x_{2\_ref}\left(t\right)\\ \left(1-\mu \right)x_{1\_ref}\left(t\right)\end{array}\right)$ (13)

The Equation (13) allows suitable choices of PV voltage and load current values to be made according to the duty cycle and set references $x_{1\_ref}\left(t\right)$, and $x_{2\_ref}\left(t\right)$. This Equation (13) ensures the stability of the nominal state system.

Let’s put the estimation error $e\left(t\right)=\left(\begin{array}{c}{e}_1\left(t\right)\\ e_2\left(t\right)\\ e_3\left(t\right)\\ e_4\left(t\right)\end{array}\right)=\left(\begin{array}{c}{x}_1\left(t\right)-{\hat{x}}_1\left(t\right)\\ x_2\left(t\right)-{\hat{x}}_2\left(t\right)\\ d_1\left(t\right)-{\hat{d}}_1\left(t\right)\\ d_2\left(t\right)-{\hat{d}}_2\left(t\right)\end{array}\right)$.

The dynamics of the estimation error are written as:

$\stackrel{\dot{}}{e}\left(t\right)=\left(\begin{array}{c}{\stackrel{\dot{}}{e}}_1\left(t\right)\\ {\stackrel{\dot{}}{e}}_2\left(t\right)\\ {\stackrel{\dot{}}{e}}_3\left(t\right)\\ {\stackrel{\dot{}}{e}}_4\left(t\right)\end{array}\right)=\left(\begin{array}{c}{\stackrel{\dot{}}{x}}_1\left(t\right)-{\stackrel{\dot{}}{\hat{x}}}_1\left(t\right)\\ {\stackrel{\dot{}}{x}}_2\left(t\right)-{\stackrel{\dot{}}{\hat{x}}}_2\left(t\right)\\ {\stackrel{\dot{}}{d}}_1\left(t\right)-{\stackrel{\dot{}}{\hat{d}}}_1\left(t\right)\\ {\stackrel{\dot{}}{d}}_2\left(t\right)-{\stackrel{\dot{}}{\hat{d}}}_2\left(t\right)\end{array}\right)$ (14)

$\stackrel{\dot{}}{e}\left(t\right)\text{=}\left(\begin{array}{c}\frac{-R_{pv}}{L_{pv}}\left(x_1\left(t\right)-{\hat{x}}_1\left(t\right)\right)-\frac{1}{L_{pv}}\left(x_2\left(t\right)-{\hat{x}}_2\left(t\right)\right)-\frac{k_1}{{\epsilon }_1^2}\left(x_1\left(t\right)-{\hat{x}}_1\left(t\right)\right)+\left(d_1-{\hat{d}}_1\right)\\ \frac{1}{C_{dc}}\left(x_1\left(t\right)-{\hat{x}}_1\left(t\right)\right)-\frac{k_3}{{\epsilon }_2^2}\left(x_2\left(t\right)-{\hat{x}}_2\left(t\right)\right)+\left(d_2-{\hat{d}}_2\right)\\ {\stackrel{\dot{}}{d}}_1-\frac{k_2}{{\epsilon }_2^2}\left(x_1\left(t\right)-{\hat{x}}_1\left(t\right)\right)\\ {\stackrel{\dot{}}{d}}_2-\frac{k_4}{{\epsilon }_4^2}\left(x_2\left(t\right)-{\hat{x}}_2\left(t\right)\right)\end{array}\right)$(15)

In state matrix form, the dynamic error is written as:

$\stackrel{\dot{}}{e}\left(t\right)=\Gamma e+Eh\left(t\right)$(16)

With

$\Gamma =\left(\begin{array}{cccc}-\frac{R_{pv}}{L_{pv}}-\frac{k_1}{{\epsilon }_1^2}& -1/L_{pv}& 1& 0\\ \frac{1}{C_{dc}}& -\frac{k_3}{{\epsilon }_3^2}& 0& 1\\ -\frac{k_2}{{\epsilon }_2^2}& 0& 0& 0\\ 0& -\frac{k_4}{{\epsilon }_4^2}& 0& 0\end{array}\right)$, $E=\left(\begin{array}{cc}0& 0\\ 0& 0\\ 1& 0\\ 0& 1\end{array}\right)$, $h\left(t\right)=\left(\begin{array}{c}{\stackrel{\dot{}}{d}}_1\left(t\right)\\ {\stackrel{\dot{}}{d}}_2\left(t\right)\end{array}\right)$

The dynamic error solution becomes:

$e\left(t\right)=\exp \left(\Gamma t\right)e\left(0\right)+Eh\left(t\right){\displaystyle \underset{0}{\overset{t}{\int }}\exp \left(\Gamma \left(t-\tau \right)\right)\text{d}\tau }$ (17)

where $e\left(0\right)$ is the initial estimate error.

The stability of the extended state system is based on the stability of the matrix $\Gamma$. Thus, the extended state system is stable if and only if $\Gamma$ is a Hurwitz matrix.

Equation (4) can be rewritten in the following form:

$\stackrel{\dot{}}{x}\left(t\right)=f\left(x\right)+g_1\left(x\right)u\left(t\right)+g_2\left(x\right)d\left(t\right)$ (18)

With, $f\left(x\right)=Ax\left(t\right)$, $g_1\left(x\right)=B$, $g_2\left(x\right)=E_d$

The estimated state equation is written:

$\stackrel{\dot{}}{\hat{x}}\left(t\right)=f\left(\hat{x}\right)+g_1\left(\hat{x}\right)u\left(t\right)+g_2\left(\hat{x}\right)\hat{d}\left(t\right)$ (19)

Either:

$s=\hat{e}+\beta {\displaystyle \int \hat{e}\text{d}t}$ (20)

The sliding surface of the integral form [42] , with $\hat{e}=\hat{x}-x_d$ being the error at the states, $\hat{x}$ the estimated states, and $x_d$ the states desired by the system control.

By deriving the sliding surface $\stackrel{\dot{}}{s}=\stackrel{\dot{}}{\hat{e}}+\beta \hat{e}=\left(\stackrel{\dot{}}{\hat{x}}-{\stackrel{\dot{}}{x}}_d\right)+\beta \left(\hat{x}-x_d\right)$ and combining it with Equation (13), we obtain:

$\stackrel{\dot{}}{s}=0\Rightarrow \left(\stackrel{\dot{}}{\hat{x}}-{\stackrel{\dot{}}{x}}_d\right)+\beta \left(\hat{x}-x_d\right)=0\Rightarrow f\left(\hat{x}\right)+g_1\left(\hat{x}\right)u\left(t\right)+g_2\left(\hat{x}\right)\widehat{\stackrel{¯}{\stackrel{¯}{d}}}\left(t\right)-{\stackrel{\dot{}}{x}}_d+\beta \hat{e}=0$

So, the control becomes:

$u^{\ast }\left(t\right)={\left(g_1\left(\hat{x}\right)\right)}^{-1}\left[{\stackrel{\dot{}}{x}}_d-\beta \hat{e}-f\left(\hat{x}\right)-g_2\left(\hat{x}\right)\cdot \hat{d}\left(t\right)-K\cdot sign\left(s\right)-s\frac{{\left|g_2\left(\hat{x}\right)\right|}^2}{\eta \left(\hat{x}\right)}\right]$ (21)

With $\eta >0$, $\beta >0$, $K>D_{\sup }$, $\hat{d}\left(t\right)$ are bounded perturbations and $\left|\hat{d}\left(t\right)\right|\le D_{\sup }$, $\hat{e}=\left(\begin{array}{c}{\hat{e}}_1\\ {\hat{e}}_2\end{array}\right)$.

$\hat{x}$ and $\hat{d}\left(t\right)$ are estimated by the standard extended state observer.

Figure 4 schematically shows the block diagram of the implementation of the standard extended state observer (SESO) and Sliding Mode Control (SMC) combination methods.

Hypothesis 3: The disturbances and their derivatives are bounded, and have constant values in steady state, so:

$\underset{t\to \infty }{\lim }\stackrel{\dot{}}{d}\left(t,x,u\right)=0$ and $\underset{t\to \infty }{\lim }d\left(t,x,u\right)=\sup \left|d\left(t,x,u\right)\right|$

Consider the following Lyapunov function:

$V=\frac{1}{2}{s}^2$ (22)

The derivative of the Lyapunov function gives:

$\stackrel{\dot{}}{V}=\stackrel{\dot{}}{s}s$ (23)

Considering Equations (18) to (21):

$\begin{array}{c}\stackrel{\dot{}}{V}=s\left(\beta \hat{e}+\left(\stackrel{\dot{}}{\hat{x}}-{\stackrel{\dot{}}{x}}_d\right)\right)\\ =s\left(\beta \hat{e}+f\left(\hat{x}\right)+g_1\left(\hat{x}\right)u\left(t\right)+g_2\left(\hat{x}\right)\hat{d}\left(t\right)-{\stackrel{\dot{}}{x}}_d\right)\end{array}$ (24)

Given that $g_1\left(\hat{x}\right)u\left(t\right)={\stackrel{\dot{}}{x}}_d-\beta \hat{e}-f\left(\hat{x}\right)-g_2\left(\hat{x}\right)\cdot \hat{d}\left(t\right)-K\cdot sign\left(s\right)-s\frac{{\left|g_2\left(\hat{x}\right)\right|}^2}{\eta \left(\hat{x}\right)}$

$\begin{array}{c}\stackrel{\dot{}}{V}=s(\beta \hat{e}+f\left(\hat{x}\right)+{\stackrel{\dot{}}{x}}_d-\beta \hat{e}-f\left(\hat{x}\right)-g_2\left(\hat{x}\right)\cdot \hat{d}\left(t\right)-K\cdot sign\left(s\right)\underset{}{\overset{}{\stackrel{}{}}}\\ -s\frac{{\left|g_2\left(\hat{x}\right)\right|}^2}{\eta \left(\hat{x}\right)}+g_2\left(\hat{x}\right)\hat{d}\left(t\right)-{\stackrel{\dot{}}{x}}_d)\\ =s\left(-K\cdot sign\left(s\right)-s\frac{{\left|g_2\left(\hat{x}\right)\right|}^2}{\eta \left(\hat{x}\right)}\right)\end{array}$ (25)

So,

$\stackrel{\dot{}}{V}\le -\left|s\right|k-{\left|s\right|}^2\frac{{\left|g_2\left(\hat{x}\right)\right|}^2}{\eta \left(\hat{x}\right)}$ (26)

Or $K>0$ and $\eta \left(\hat{x}\right)>0$, so for everything $s\ne 0$:

$\stackrel{\dot{}}{V}<0$ (27)

Which means $\underset{t\to \infty }{\lim }\left(s\right)\approx 0$.

The sliding surface is thus reached, and steady-state stability is ensured by the sliding surface (22).

The mixed of the sliding mode control (SMC) and standard extended state observer (SESO) approaches applied to the Photovoltaic system associated with a boost converter and DC load, subject to mismatched disturbances, has been implemented and validated by Matlab/Simulink software according to the parameters shown in Table 1 . The Boost parameters, PV voltage, and DC bus voltage are taken from work done by Palak Jain et al. [43] .

By correctly choosing the parameters of the standard extended state observer of Equation (15), such as: ${\epsilon }_1=\text{0}\text{.0065}$ ${\epsilon }_2=0.0095$ and $E_1=E_2=\left(\begin{array}{cc}-3& 1\\ -1& 0\end{array}\right)$.

The eigenvalues of $E_1$ and $E_2$ are $\left\{-2,6180;-0,3820\right\}$. These eigenvalues are strictly negative real decimal numbers.

Thus, the solutions of the standard extended state observer (Equation (10)) are stable.

The sliding mode control parameters are as follows:

$K=\left[\begin{array}{c}{K}_1\\ K_2\end{array}\right]=\left[\begin{array}{c}40000\\ 19100\end{array}\right]>\left[\begin{array}{c}\left|V_{pv}/L_{pv}\right|\\ \left|-I_{load}/C_{dc}\right|\end{array}\right]$, $\beta =\left(\begin{array}{c}30\\ 30\end{array}\right)$, $\eta \left(x\right)=\left(\begin{array}{c}{\eta }_1\\ {\eta }_2\end{array}\right)=\left(\begin{array}{c}2\\ 1\end{array}\right)$,

$c_{01}=5\times {10}^{-6}$, $s_{01}=e_{01}$, $c_{02}=5\times {10}^{-6}$, $s_{02}=e_{02}$, $s=\left(\begin{array}{c}{s}_1\\ s_2\end{array}\right)=\left(\begin{array}{c}{c}_{01}\cdot s_{01}+e_1\\ c_{02}\cdot s_{02}+e_2\end{array}\right)$

Figure 5 illustrates the actual inductance current alongside its estimated value generated by the combination of SMC and SESO. The estimated error ranges from 0.0 A to 0.5 A, resulting in a relative error of 4%. This indicates that the estimated values are close to the actual values. The results demonstrate that the estimator accurately predicts the current, allowing effective tracking of the set reference, even when the current peaks around 190 A at the beginning of the simulation.

Figure 6 displays the actual DC bus voltage estimated by combining SMC and SESO. From the estimated error curve, it tends to almost zero. This indicates that the estimated values are close to the actual values. The results demonstrate that the estimator accurately predicts the voltage of the DC bus.

Figure 7 shows the real and estimated d1 disturbances of the system. The error curve tends to zero, with an order of 10⁻⁵. The estimated values obtained by the combination of SESO and SMC follow the real values of the disturbances d1.

Figure 8 shows the studied system’s real and estimated d2 disturbances. The error curve tends to zero, with an order of 10⁻⁶. The values estimated by the combination of SESO and SMC align with the actual disturbances’ values.

This section tests the robustness of the SESO and SMC combination methods against PV voltage and DC load disturbances.

In this work, it is assumed that the voltage $V_{pv}$ is perturbed due to the clouds or for other mismatches and changes as follows:

$V_{pv}=\{\begin{array}{l}240\text{V, for}t\in \left[0;0.5\right[\mathrm{s}\\ 220\text{V, for}t\in \left[0.5;1\right[\mathrm{s}\\ 230\text{V, for}t\in \left[1;2\right[\mathrm{s}\\ 250\text{V, for}t\in \left[2;3\right[\mathrm{s}\\ \text{240 V, for}t\in \left[3;5\right]\mathrm{s}\end{array}$ (28)

Figure 9 illustrates the actual inductance current and estimated value generated by the combination of SMC and SESO, in the presence of external PV voltage perturbations. At the beginning of the variation time (0.5 s), the current is subject to oscillations (of small amplitude). The small oscillations of the estimated error range from 0.0 A to 0.5 A at all variation times (0.5 s, 1 s, 2 s, 3 s), resulting in a maximum relative error of 4%. This indicates that the estimated values are close to the actual values. The results demonstrate that the combined methods accurately predict the current, allowing effective tracking of the set reference, although PV voltage varies from the set reference (variation of maximum 20 V). So, the combined methods are robust to PV voltage variation while estimating inductance current.

Figure 10 displays the actual DC bus voltage estimated by combining SMC and SESO, in the presence of external PV voltage perturbations. From the estimated error curve, it tends to almost zero. The DC bus voltage is subject to oscillations (of tiny amplitude) while the PV voltage varies. The small oscillations of the estimated error are quietly zero, of the order of 10⁻⁴. This indicates that the estimated values are close to the actual values. The results demonstrate that combined methods accurately predict the current, allowing effective tracking of the set reference, although PV voltage varies from the set reference (variation of maximum 20 V). So, the combined methods are robust to PV voltage variation while estimating DC bus voltage.

Figure 11 shows the system’s real and estimated d1 disturbances in the presence of external PV voltage perturbations. The estimated d1 curve does not follow the real curve precisely. It takes the shape of the real curve but does not have closed values. The error curve reveals a gap between the real and estimated disturbances d1, with an order of 10³. When PV voltage varies, the estimated values obtained by combining SESO and SMC do not follow the real values of the disturbances d1, even though they take the form, resulting in a maximum relative error of 20%. So, the combined methods are not robust to PV voltage variation while estimating disturbances d1.

Figure 12 shows the system’s real and estimated d2 disturbances in the presence of external PV voltage perturbations. The estimated d2 curve follows the real curve. The disturbances d2 are subject to oscillations (of considerable amplitude) while the PV voltage varies. The error curve reveals a gap between the real and estimated disturbances d1, with an order of 100. When PV voltage varies, the estimated values obtained by combining SESO and SMC do not follow the real values of the disturbances d2, even though the real values oscillate with a considerable amplitude, resulting in a maximum relative error of 18%. So, the combined methods are not robust to PV voltage variation while estimating disturbances d2.

In this work, it is assumed that the DC load current $I_{load}$ is perturbed, and changes as follows:

$I_{load}=\{\begin{array}{l}12\text{A, for}t\in \left[0;0.5\right[\mathrm{s}\\ 10\text{A, for}t\in \left[0.5;2\right[\mathrm{s}\\ 8\text{A, for}t\in \left[2;3\right]\mathrm{s}\\ 12\text{A, for}t\in \left[3;5\right]\mathrm{s}\end{array}$ (29)

Figure 13 illustrates the actual inductance current and estimated value generated by the combination of SMC and SESO, in the presence of external load current perturbations. At the beginning of the variation time (0.5 s), the current is subject to oscillations (of small amplitude). The small oscillations of the estimated error range from 0.0 A to 0.5 A at all variation times (0.5 s, 1 s, 2 s, 3 s), resulting in a maximum relative error of 4%. This indicates that the estimated values are close to the actual values. The results demonstrate that the combined methods accurately predict the current, allowing effective tracking of the set reference, although DC load current varies from the set reference (variation of maximum 4 A). So, the combined methods are robust to DC load current variation while estimating inductance current.

Figure 14 displays the actual DC bus voltage estimated by combining SMC and SESO, in the presence of external DC load current perturbations. From the estimated error curve, it tends to almost zero. The DC bus voltage is subject to oscillations (of tiny amplitude) while the DC load current varies. The small oscillations of the estimated error are quietly zero, of the order of 10⁻⁴. This indicates that the estimated values are close to the actual values. The results demonstrate that combined methods accurately predict the current, allowing effective tracking of the set reference, although DC load current varies from the set reference (variation of maximum 4 A). So, the combined methods are robust to DC load current variation while estimating DC bus voltage.

Figure 15 shows the system’s real and estimated d1 disturbances in the presence of external DC load perturbations. The estimated d1 curve does not follow the real curve precisely. It does not take the form of the real curve. The error curve reveals a gap between the real and estimated disturbances d1, with an order of 2 × 10³. When DC load current varies, the estimated values obtained by combining SESO and SMC do not follow the real values of the disturbances d1, resulting in a relative error of over 20%. So, the combined methods are not robust to DC load current variation while estimating disturbances d1.

Figure 16 shows the system’s real and estimated d2 disturbances in the presence of external PV voltage perturbations. The estimated d2 curve follows the real curve. The disturbances d2 are subject to oscillations (of considerable amplitude) while the PV voltage varies. The error curve reveals a gap between the real and estimated disturbances d1, with an order of 200. When DC load current varies, the estimated values obtained by combining SESO and SMC follow the real values of the disturbances d2, even though the real values oscillate with a considerable amplitude, resulting in a relative error of over 30%. So, the combined methods are not robust to DC load current variation while estimating disturbances d2.

The chattering phenomenon persists in the control law. This is due to the sign (.) function. To mitigate this high-frequency disturbance, several previous works were drawn upon [44] . This allowed us to replace the sign (.) function with the hyperbolic tangent tanh (.). It has also been observed that increasing the SMC parameter K reduces the chattering phenomenon in the control law.

Table 2 presents the quantitative performance indicators for the combination of the sliding mode control (SMC) and standard extended state observer (SESO) methods in estimating the states of a Photovoltaic system associated with a boost converter and DC load, subject to mismatched disturbances. This table explains and compares the performance indicators for each state in the absence and the presence of perturbations. Globally, these performance indicators show good robustness for the principal states (IL and Vbus_DC) and poor disturbance reconstruction for lumped state disturbances.