H ∞ fuzzy proportional integral state feedback controller of photovoltaic systems under asymmetric actuator constraints

Abstract

This paper presents a new strategy for a robust maximum power point (MPP) tracking fuzzy controller for photovoltaic (PV) systems subject to actuator asymmetric saturation. A DC-DC boost converter is used to connect a PV panel with an output load. The output voltage of the DC-DC boost converter can be adjusted by duty ratio that is limited between 0 and 1. The aim of our control design is to track the MPP under atmospheric condition changes and the presence of the asymmetric saturation of the duty ratio. To minimize tracking error and disturbance effect, the dynamic behaviour of a PV system and its reference model are described by using Takagi–Sugeno fuzzy models. Then, a constrained control based on a fuzzy PI state feedback controller is proposed. The H_∞ control approach is used in control design and stability conditions of the closed-loop system are formulated and solved in terms of linear matrix inequalities. Finally, simulation results are given to show the tracking performance of the control design.

Keywords

Asymmetric saturation photovoltaic system DC-DC boost converter maximum power point tracking Takagi–Sugeno fuzzy model linear matrix inequalities

Introduction

In order to satisfy our increasing energy needs, big challenges should be undertaken in the future. Renewable energies such as fuel cells (Cossutta et al., 2015), biomass plants (Singh and Subhash, 2016), turbines generators (Meghni et al., 2018; Rahim and Khan, 2014) and photovoltaic (PV) arrays (Azri et al., 2014; Vincheh et al., 2014) are inexhaustible sources of energy. The PV effect has the advantage to directly transform sunlight into electrical energy using a DC-DC converter to adjust the PV input voltage by means of variation of duty ratio (Brunton et al., 2010; Mastromauro et al., 2009). PV systems have strong nonlinear characteristics and the output power varies with climatic conditions. As a result, the control design of PV systems is a very difficult task, especially with unpredictable atmospheric conditions.

In this context, many approaches have been proposed to maximize the PV power output, such as the perturb-and-observe (P&O) method (Abdessalem et al., 2011; De Brito et al., 2013; Lyden and Haque, 2015; Subudhi and Pradhan, 2012), incremental conductance (INC) methods (Abdessalem et al., 2011; Liu et al., 2008; Sera et al., 2013; Vahid et al., 2018), the maximum power point (MPP) estimation method (Abdessalem et al., 2011; Mohammed et al., 2002) and artificial intelligence methods (Chiu, 2010; Chiu and Ouyang, 2011; Dahmane et al., 2013; Nabulsi and Dhaouadi, 2012; Salah and Ouali, 2011). Vahid et al. (2018) have proposed a robust feedback linearization controller with a modified INC method for MPP tracking (MPPT) in the PV systems and the overall closed-loop internal stability is guaranteed. However, the system arrives at the MPP only in acceptable time. Nabulsi and Dhaouadi (2012) have proposed a new fuzzy logic controller (FLC)-based P&O scheme in which the PV reference voltage is self-adjusted with variable step size by which the MPP is accurately reached. As well, the INC method is influenced by the same factors as the P&O method implicating a compromise between tracking speed and oscillations to reach a MPP. Open-circuit voltage and short-circuit current (SCC) algorithms proposed by Masoum et al. (2002) are unable to find the true MPP due to the approximation used in these algorithms. In the study by Priyabrata and Priyabrat (2019), an analog circuitry-based fast and robust MPPT method utilizing a boost DC-DC converter is presented to improve the tracking capability.

The P&O and INC methods are widely used in commercial PV panels due to their simplicity, easy implementation and low cost. However, in these traditional controls, a PV system with a DC-DC boost converter is considered as linear system, which results in poor dynamic performance and considerable oscillations around the MPP. In addition, the external disturbance is not taken into account, which results in a slow and a wrong tracking during rapid change of atmospheric conditions. On the other hand, in these classical approaches, the problem of the presence of the asymmetric saturation of duty ratio (limited between 0 and 1) is not treated, which is a source of degradation of performances. Indeed, actuator saturation can degrade the performance of a closed-loop system and sometimes destabilize a stable closed-loop system (Cao and Lin, 2003b; Henrion, 1999).

Many stabilization conditions results for nonlinear systems subject to actuator saturation have been reported in the literature and one can distinguish two approaches. In one approach, the control objective is to avoid the saturation limits (Cao and Lin, 2003a, 2003b; Han, 2007; Nasri et al., 2019; Saifia et al., 2010b, 2011, 2012a, 2012b, 2012c, 2012d, 2019, 2020). In the other approach, an estimation of the domain of attraction in which any initialization of the system states does not lead to instability in the presence of saturation is carried out (Cao and Lin, 2003a, 2003b; Nasri et al., 2019; Saifia et al. 2010a, 2010b, 2011, 2012a, 2012b, 2012c, 2012d, 2019, 2020).

In this work, the well-known Takagi–Sugeno (T-S) fuzzy models are considered as a useful tool for approximating complex nonlinear systems (Meghni et al., 2018; Wang et al., 1996). The fuzzy model proposed by Takagi and Sugeno (1985) is described by fuzzy IF-THEN rules which represent local input–output relations of a nonlinear system. The main feature of a T-S fuzzy model is its ability to express the local dynamics of each fuzzy implication (rule) by a linear system model. The overall fuzzy model of the system is achieved by fuzzy ‘blending’ of the linear system models. By using a parallel distributed compensation law, the local linear models allow application of the linear control theory to nonlinear complex systems. The main interest is that the controller’s gain can be calculated from the stability conditions of the augmented T-S fuzzy system, which can be easily transformed into linear matrix inequalities (LMIs) and solved efficiently by convex programming techniques (Singh and Subhash, 2016; Vincheh et al., 2014).

To design controllers for nonlinear systems, many LMI stabilization conditions have been derived based on T-S fuzzy models via the Lyapunov approach. Estrada-Manzo et al. (2019) present LMI-based relaxed stabilization conditions of T-S discrete fuzzy models via static output feedback controller and Lyapunov function. The problem of robust H_∞ control for a class of fuzzy time-delay systems has been investigated in Xin et al. (2019) by introducing the Wirtinger-type integral inequality and convex technique to estimate the derivative of Lyapunov–Krasovskii functional, which guarantees robust asymptotic stability and a prescribed H_∞ performance level of the corresponding closed-loop system. Lien et al. (2019) consider the mixed H₂/H_∞ performance for a class of uncertain T-S fuzzy systems with time delays and linear fractional perturbations based on the LMI optimization approach. In Saifia et al. (2020), a robust H_∞ static output-feedback controller for discrete T-S fuzzy models subject to input saturation constraint has been developed and the stabilization conditions of the fuzzy system are formulated as a convex optimization problem in terms of LMIs. In particular, a TS model-based method has been used for MPPT of PV systems. Chiu and Ouyang (2011) design a robust observed-based MPPT control for PV systems with DC-DC buck converter by using a Lyapunov approach and LMI formulation. Interesting results for maximizing power point tracking of PV systems have been reported by Allouche et al. (2018) who propose a robust-MPPT fuzzy controller of PV system to guarantee both H₂ optimal control and H_∞ model. Hafedh et al. (2012) display an intelligent control strategy based on the T-S representation for the MPPT of PV system; the optimal duty cycle is computed based on the T-S fuzzy system, which permits the maximum power to be extracted from the PV array panel, showing the advantage of the suggested MPPT routine above classical configuration. Alongside that, Khabou et al. (2020) deploy a control technique for the MPPT of a PV system under atmospheric conditions changes; a robust-MPPT fuzzy controller of PV system has then investigated which registered fast convergence to the maximum power and elimination of the oscillations around the maximum and the robustness. In Ajaamoum et al. (2015), a technique for improving and optimizing the performances of PV system with DC-DC buck converter has been introduced based on the T-S fuzzy system. However, in these works, authors have not considered the problem of actuator saturation which can destabilize the closed-loop system.

For this purpose and to the best of our knowledge, a T-S fuzzy proportional integral state feedback control has never been applied to PV systems with DC-DC boost converter under actuators asymmetric saturation and disturbance effect. To avoid this drawback, following contributions are proposed:

A new strategy for maximizing power point tracking of PV systems based on TS fuzzy proportional integral state feedback and an H_∞ performance is proposed.

New H_∞ stabilization conditions of PV systems with DC-DC boost converter under rapid change of atmospheric conditions are derived and formulated in terms of a LMI.

The PV systems with DC-DC boost converter input constraint is taken into consideration and the duty ratio asymmetric saturation is transformed in a symmetric saturation between −0.5 and 0.5 using mathematical transformation.

This paper is organized as follows. The next section gives a TS fuzzy representation of a PV system with a DC-DC boost converter. Then, H_∞ stabilization conditions of the MPPT controller subject to asymmetric actuator saturation and external disturbances based on fuzzy proportional integral state feedback are derived and formulated in terms of LMIs. Finally, simulation results are given to show the effectiveness of the proposed fuzzy control approach.

The notation used is as follows: $i \in I_{m}$ is used to denote $i \in {1, . . ., m}$ , $R^{n \times m}$ is the set of real matrices. The notation $M > (<)$ is used to denote a symmetric positive definite (negative definite) matrix; $(*)$ indicates a transpose quantity in a symmetric matrix.

Fuzzy modelling of PV generator systems

In this work, a PV system is composed of a PV panel and a DC-DC boost converter. Figure 1 illustrates the system and in the following sections each component is described.

Photovoltaic generator model

The PV cell is modelled as a p-n semiconductor junction that directly transforms light energy into electricity; it is composed of solar cells arranged in an np-parallel, ns-series configuration. The equivalent circuit of a PV cell based on a single diode is presented in Figure 2, where $I_{pv}$ , and $V_{pv}$ , respectively, indicate the output current and voltage of the PV array. We assume that the internal shunt and series resistances are neglected. The output current of the PV array is given by the following analytic expression:

I_{pv} = n_{p} I_{ph} - n_{p} I_{0} [\exp (\frac{K_{pv} V_{pv}}{n_{s}}) - 1]

(1)

K_{pv} = \frac{q}{ATK}

(2)

where $T$ is the cell temperature, $q = 1.6 \times 10^{- 19} C$ is the electronic charge, $K = 1.3805 \times 10^{- 23} j / K$ is the Boltzmann constant, the ideal p-n junction characteristic factor $A = 1 - 5$ , $I_{ph}$ is the light-generated current and $I_{0}$ is the reverse saturation current. Moreover, the reverse saturation current and the light-generated current influenced by insolation and temperature with the following expressions (Makhloufi, 2014):

I_{ph} = [I_{phR} - μ_{Icc} (T - T_{R})] \frac{λ}{λ_{R}}

(3)

I_{0} = I_{0 R} {(\frac{T}{T_{R}})}^{3} \exp [\frac{q E_{gap}}{AK} (\frac{1}{T_{R}} - \frac{1}{T})]

(4)

where $I_{phR}$ is the short-circuit cell current at reference condition $(T_{R}, λ_{R})$ , $μ_{Icc}$ (in milliamperes per kelvin) is the SCC temperature coefficient, $λ$ is the insolation (in milliwatts per square centimetre), $I_{OR}$ is the reverse saturation current at the reference condition $(T_{R}, λ_{R})$ and $E_{gap} = 1.12 eV$ is the bandgap energy of the semiconductor making up the cell. The power–voltage and power–current characteristics illustrated in Figure 3 show the characteristics curve of the power related to the PV voltage and current for different values of solar irradiation $λ (W / m^{2})$ and temperature $T$ . It can be seen that the maximum power curve changes with insolation and/or cell temperature. Furthermore, for each power curve there exists only one point such as the generated PV power is maximized.

DC-DC boost converter

A dc-dc boost converter is connected to the PV array to adjust the PV array power, as shown in Figure 1. The dynamic model of the converter can be defined by state space equations (Allouche et al., 2018), where the panel voltage $V_{pv} (t)$ , the inductor current $i_{L} (t)$ and the load voltage $V_{0} (t)$ are taken as state variables, as follows:

{\begin{matrix} {\overset{\cdot}{V}}_{pv} = - \frac{1}{C_{a}} i_{L} + \frac{1}{C_{a}} i_{pv} \\ {\overset{\cdot}{i}}_{L} = \frac{1}{L} V_{pv} - \frac{R_{L}}{L} i_{L} - \frac{1}{L} V_{0} (1 - U) \\ {\overset{\cdot}{V}}_{0} = \frac{1}{C_{b}} i_{L} (1 - U) - \frac{1}{R_{0} C_{b}} V_{0} \end{matrix}

(5)

$V_{pv}$ represents the PV array voltage on the capacitance $C_{a}$ , $i_{L}$ represents the current on the inductance $L$ and $V_{0}$ the voltage on the capacitance $C_{b}$ , $U (t) \in [0, 1]$ is the duty ratio of the pulse width-modulated signal to control the power switcher, $R_{L}$ is the internal resistance of the inductance $L$ . Equation (5) can be described by the following compact nonlinear model:

{\overset{\cdot}{x}}_{st} (t) = A x_{st} (t) + B (x (t)) U (t) + G i_{pv} (t)

(6)

with

\begin{matrix} A = (\begin{matrix} 0 & - \frac{1}{C_{a}} & 0 \\ \frac{1}{L} & - \frac{R_{L}}{L} & - \frac{1}{L} \\ 0 & \frac{1}{C_{b}} & - \frac{1}{R_{0} C_{b}} \end{matrix}), B (x (t)) = (\begin{matrix} 0 \\ \frac{V_{0}}{L} \\ - \frac{i_{L}}{C_{b}} \end{matrix}), \\ G = (\begin{matrix} \frac{1}{Ca} \\ 0 \\ 0 \end{matrix}), x_{st} (t) = (\begin{matrix} V_{pv} \\ i_{L} \\ V_{0} \end{matrix}) \end{matrix}

T-S fuzzy model of the PV system

In this section, we will use a T-S fuzzy model to represent the solar power generation system with the DC-DC boost converter. Several studies have proved that the T-S fuzzy model can describe the behaviour of continuous nonlinear systems by combining local linear dynamic subsystems in IF–THEN fuzzy rules. The PV nonlinear model (equation (6)) is transformed into a T–S fuzzy model using sector nonlinearity transformation and assuming that the inductor current $i_{L} (t)$ and the load voltage $V_{0} (t)$ as decision variables.

{\begin{matrix} η_{1} (t) = V_{0} (t) \\ η_{2} (t) = i_{L} (t) \end{matrix}

(7)

Supposing that the decision variables $V_{0} (t)$ and $i_{L} (t)$ are bounded as:

{\begin{matrix} η_{1, min} \leq η_{1} (t) \leq η_{1, max} \\ η_{2, min} \leq η_{2} (t) \leq η_{2, max} \end{matrix}

(8)

The nonlinear system (equation (6)) can be described by a T–S model with $r = 2^{n} = 2^{2}$ fuzzy if–then rules, as follows:

Rule i:

if η_{1} (t) is M_{1, a} and η_{2} (t) is M_{2, b} Then

{\overset{\cdot}{x}}_{st} (t) = A x_{st} (t) + B_{i} u (t) + G i_{pv} (t)

(9)

where $a, b \in {min, max}$

\begin{matrix} B_{1} = (\begin{matrix} 0 \\ \frac{V_{0 min} (t)}{L} \\ - \frac{i_{L min} (t)}{C_{b}} \end{matrix}), B_{2} = (\begin{matrix} 0 \\ \frac{V_{0 min} (t)}{L} \\ - \frac{i_{L max} (t)}{C_{b}} \end{matrix}), \\ B_{3} = (\begin{matrix} 0 \\ \frac{V_{0 max} (t)}{L} \\ - \frac{i_{L min} (t)}{C_{b}} \end{matrix}), B_{4} = (\begin{matrix} 0 \\ \frac{V_{0 max} (t)}{L} \\ - \frac{i_{L max} (t)}{C_{b}} \end{matrix}) \end{matrix}

The global fuzzy model is given by:

{\overset{\cdot}{x}}_{st} (t) = \sum_{i = 1}^{r} λ_{i} (η (t)) (A x_{st} (t) + B_{i} u (t) + G i_{pv} (t))

(10)

where $λ_{i} (η) = ω_{i} (η) / \sum_{i = 1}^{r} ω_{i} (η)$ , $ω_{i} (η) = M_{1, a} M_{2, b}$ with $a, b \in {min, max}$ , for all $t > 0$ , $λ_{i} (η) \geq 0$ and $\sum_{i = 1}^{r} λ_{i} (η) = 1$ :

{\begin{matrix} M_{i, min} (η_{i} (t)) = \frac{η_{i} (t) - η_{i, min}}{η_{i, max} - η_{i, min}} \\ M_{i, max} (η_{i} (t)) = \frac{η_{i, max} - η_{i} (t)}{η_{i, max} - η_{i, min}} = 1 - M_{i, min} (η_{i} (t)) \end{matrix}

Robust T-S fuzzy MPPT control design under asymmetric input constraint

In this section, we present a robust T-S fuzzy MPPT control design under asymmetric constraint of duty ratio. Indeed, the PV effect directly transforms solar energy into electrical energy using a DC-DC converter to adjust the PV input voltage by means of variations of duty ratio which is subject to saturation between 0 and 1 (Saifia et al., 2012b).

It is well known that actuator saturation can degrade the closed-loop system performance or destabilize a stable closed-loop system (Cao and Lin, 2003b; Da Silva et al., 1997; Henrion, 1999). To consider this problem, in this work, a mathematical transformation is used to represent the asymmetric constraint as a symmetric saturation between −0.5 and 0.5.

Reference model

To track the MPP, we have to drive the error $(x_{st} (t) - x_{ref} (t))$ to zero. For that we design the MPP reference model based on the T-S fuzzy model structure of the DC-DC boost converter to generate the desired reference state to be tracked.

We define the state model of the MPP reference model as follows:

{\overset{\cdot}{x}}_{ref} (t) = A_{ref} x_{ref} (t) + G_{ref} I_{pvOp}

(11)

with

\begin{matrix} A_{ref} = (\begin{matrix} 0 & - \frac{1}{C_{a}} & 0 \\ \frac{1}{L} & - \frac{R_{L}}{L} & - \frac{1}{L} (1 - U_{op}) \\ 0 & \frac{1}{C_{b}} (1 - U_{op}) & - \frac{1}{R_{0} C_{b}} \end{matrix}), \\ G_{ref} = (\begin{matrix} \frac{1}{Ca} \\ 0 \\ 0 \end{matrix}), U_{op} = 1 - \sqrt{\frac{V_{pvOp}}{R_{0} I_{pvOp}}} \end{matrix}

From equation (11), we can note that the reference model is also nonlinear via the premise variable $z_{Re f} = (1 - U_{op})$ . Using the sector nonlinearity approach, we get the global T-S fuzzy reference model as follows:

{\overset{\cdot}{x}}_{ref} (t) = \sum_{l = 1}^{2} λ_{l} (η (t)) (A_{refl} x_{Re f} (t) + G_{ref} I_{pvOp})

(12)

with

\begin{matrix} A_{ref 1} = (\begin{matrix} 0 & - \frac{1}{C_{a}} & 0 \\ \frac{1}{L} & - \frac{R_{L}}{L} & - \frac{1}{L} z_{r, min} \\ 0 & \frac{1}{C_{b}} z_{r, min} & - \frac{1}{R_{0} C_{b}} \end{matrix}), \\ A_{ref 2} = (\begin{matrix} 0 & - \frac{1}{C_{a}} & 0 \\ \frac{1}{L} & - \frac{R_{L}}{L} & - \frac{1}{L} z_{r, max} \\ 0 & \frac{1}{C_{b}} z_{r, max} & - \frac{1}{R_{0} C_{b}} \end{matrix}), \end{matrix}

Location of MPP (V_pvOp, I_pvOp)

To reach a MPP, we must have the following condition:

\frac{d}{d i_{pv}} (V_{pv} i_{pv}) = 0

Hence,

V_{p v} + i_{p v} \frac{d V_{p v}}{d i_{p v}} = 0

(13)

Let consider now $V_{pvOp}$ and $I_{pvOp}$ are respectively the values of the optimal current and the optimal voltage when the output power of the PV panel reaches the maximum value.

By substituting $V_{pvOp}$ and $\frac{d V_{pv}}{d i_{pv}}$ by their values in equation (13), we obtain the following equation (Ohtake et al., 2003):

I_{pvOp} - (I_{pvOp} - I_{ph} + I_{0}) \ln (\frac{I_{pvOp} - I_{ph} + I_{0}}{I_{0}}) = 0

(14)

By resolving equation (14), we obtain a linear relationship between $I_{pvOp}$ , the PV current corresponding to maximum power, and $I_{ph}$ , the cell SCC.

I_{pvOp} = 0.94 I_{ph}

(15)

H_∞ tracking control design under asymmetric input constraint

The main objective of the control robustness design is to achieve the MPPT control considering an unknown disturbance $w (t)$ , this disturbance being related to climatic variations. We use the H_∞ performance criterion to design a robust controller for a PV system under asymmetric input constraint. We consider the solar power generation system (equation (6)) as follows:

{\overset{\cdot}{x}}_{st} (t) = \sum_{i = 1}^{r} λ_{i} (η (t)) [A x_{st} (t) + B_{i} u (t) + G w (t)]

(16)

where $w (t) = i_{pv} (t)$ .

The global structure of the controller is shown in Figure 4. The idea is to force the PV system to track a desired trajectory which is provided by a reference model.

Let us consider the scalar function $λ_{i} (η (t))$ , the matrix $A_{ref - l}$ and the vector $B_{i}$ for $l \in {1, . . ., r_{ref}}$ and $i \in {1, . . ., r}$ with appropriate dimensions. We will denote:

B_{η} = \sum_{i = 1}^{r} λ_{i} (η (t)) B_{i}, A_{η ref} = \sum_{l = 1}^{r_{ref}} λ_{l} (η (t)) A_{refl} .

and let

err (t) = x_{st} (t) - x_{ref} (t)

(17)

From equations (12), (15) and (17), we can obtain the fuzzy augmented system as follows:

\overset{\cdot}{\tilde{X}} = {\tilde{A}}_{η} \tilde{X} + {\tilde{B}}_{η} U + \tilde{G} \tilde{w}

(18)

where

\begin{matrix} \tilde{X} = (\begin{matrix} x_{st} \\ x_{ref} \\ \int err \end{matrix}), {\tilde{A}}_{η} = (\begin{matrix} A^{η} & O & O \\ O & A_{η ref} & O \\ 1 & - 1 & O \end{matrix}), \\ {\tilde{B}}_{η} = (\begin{matrix} B_{η} \\ O \\ O \end{matrix}), \tilde{G} = (\begin{matrix} G & O \\ O & G_{ref} \\ O & O \end{matrix}), \tilde{w} = (\begin{matrix} w (t) \\ I_{pvOp} \end{matrix}) \end{matrix}

where $O$ is a zero matrix of appropriate dimension.

The control input (duty-cycle) is a constraint in amplitude with the following characteristics:

0 \leq U (t) \leq 1

(19)

which is an asymmetric saturation. The system described in equation (18) can be written as follows:

\begin{matrix} \overset{\cdot}{\tilde{X}} = {\tilde{A}}_{η} \tilde{X} + {\tilde{B}}_{η} U + \tilde{G} \tilde{w} \\ \overset{\cdot}{\tilde{X}} = {\tilde{A}}_{η} \tilde{X} + \tilde{G} \tilde{w} + {\tilde{B}}_{η} (U (t) - \frac{1}{2}) + \frac{1}{2} {\tilde{B}}_{η} \\ \overset{\cdot}{\tilde{X}} (t) = {\tilde{A}}_{η} \tilde{X} + \tilde{G} \tilde{w} + 0.5 {\tilde{B}}_{η} + {\tilde{B}}_{η} U_{δ} (t) \end{matrix}

(20)

where

U_{δ} (t) = U (t) - 0.5

(21)

Therefore, the new control input $U_{δ} (t)$ is considered as symmetric saturation as:

- 0.5 \leq U_{δ} (t) \leq + 0.5

(22)

The proposed T-S fuzzy controller is based on H_∞ fuzzy proportional integral state feedback where the integral of the tracking error is introduced to reduce the tracking error. Then, the structure of T-S fuzzy controller is given as follows:

U_{δ} (t) = \sum_{j = 1}^{r} λ_{j} (η (t)) [K_{j} err (t) + L_{j} \int err (t) dt]

(23)

where $K_{j}$ and $L_{j}$ are the fuzzy control gains to be designed.

From equations (23) and (17), we can write the structure of the T-S fuzzy controller as follows:

U_{δ} (t) = \sum_{j = 1}^{4} λ_{j} (η (t)) [K_{j} x_{st} (t) - K_{j} x_{ref} (t) + L_{j} \int err (t) dt]

(24)

assuming that

\begin{matrix} K_{η} = \sum_{j = 1}^{4} λ_{j} (η (t)) K_{j}; L_{η} = \sum_{j = 1}^{4} λ_{j} (η (t)) L_{j} \\ U_{δ} (t) = (\begin{matrix} K_{η} & - K_{η} & L_{η} \end{matrix}) (\begin{matrix} x_{st} (t) \\ x_{ref} (t) \\ \int err (t) \end{matrix}) \\ U_{δ} (t) = {\tilde{K}}_{η} \tilde{X} \end{matrix}

(25)

In this work, to address the duty ratio saturation problem, we use so-called constrained control. In this case, the input control cannot be saturated, and the saturation function $U_{δ} (t)$ can be represented by equation (25) such as $\tilde{X}$ belonging to the polyhedral:

Ω \overset{Δ}{=} {\tilde{X} \in ℜ^{n} | {\tilde{K}}_{η} \tilde{X} | \leq \bar{U}}

(26)

with $\bar{U} = 0.5$ is the saturation level function.

After constructing the fuzzy augmented system (equation (20)), we introduce the structure of the T-S fuzzy controller (equation (25)) as follows:

\begin{matrix} \overset{\cdot}{\tilde{X}} (t) = {\tilde{A}}_{η} \tilde{X} + \tilde{G} \tilde{w} + 0.5 {\tilde{B}}_{η} + {\tilde{B}}_{η} U_{δ} (t) \\ \overset{\cdot}{\tilde{X}} (t) = {\tilde{A}}_{η} \tilde{X} + \tilde{G} \tilde{w} + 0.5 {\tilde{B}}_{η} + {\tilde{B}}_{η} {\tilde{K}}_{η} \tilde{X} \\ \overset{\cdot}{\tilde{X}} (t) = ({\tilde{A}}_{η} + {\tilde{B}}_{η} {\tilde{K}}_{η}) \tilde{X} + \tilde{G} \tilde{w} + 0.5 {\tilde{B}}_{η} \\ \overset{\cdot}{\tilde{X}} = {\bar{A}}_{η} \tilde{X} + \bar{G} \bar{w} \end{matrix}

(27)

with

\begin{matrix} {\bar{A}}_{η} = {\tilde{A}}_{η} + {\tilde{B}}_{η} {\tilde{K}}_{η} \\ \bar{G} = (\begin{matrix} G & O \\ O & G_{ref} \\ O & O \end{matrix} \begin{matrix} α {\tilde{B}}_{η} \\ O \\ O \end{matrix}), \\ \bar{w} = (\begin{matrix} \tilde{w} (t) \\ I \end{matrix}), \\ α = 0.5 \end{matrix}

Based on the fuzzy augmented system (equation (27)), the robust $H_{\infty}$ tracking performance and Lyapunov approaches will be used to compute the controller gains.

Let us consider the $H_{\infty}$ performance related to the tracking error given by the objective function:

J = \overset{\cdot}{v} (t) + z^{T} (t) z (t) - γ^{2} {\bar{w}}^{T} (t) \bar{w} (t) < 0

(28)

where

z (t) = e (t) = {\tilde{C}}_{e} \tilde{X} (t), {\tilde{C}}_{e} = (\begin{matrix} I & - I & O \end{matrix})

(29)

v (t) = {\tilde{X}}^{T} (t) P \tilde{X} (t)

(30)

where $v (t)$ is a quadratic Lyapunov function candidate and $P$ is a symmetric positives defined matrix.

Define an ellipsoid as:

ε (P, ρ) = {\tilde{X} \in ℜ^{3 \times n} | {\tilde{X}}^{T} (t) P \tilde{X} (t) \leq ρ, ρ > 1}

For all $e (0) \in ε (P, 1) \subset ε (P, ρ)$ , if $J ≺ 0$ , then,

\begin{matrix} \forall t_{f} > 0 : \int_{0}^{T} J dt = \int_{0}^{T} (\overset{\cdot}{v} (t) + z^{T} (t) z (t) - γ^{2} {\bar{w}}^{T} (t) \bar{w} (t)) < 0, such as : ‖ \bar{w} (t) ‖_{2}^{2} < ϖ^{2} . \end{matrix}

where $‖ \bar{w} (t) ‖_{2} = \sqrt{\int_{t = 0}^{t = \infty} {\bar{w}}^{T} (t) \bar{w} (t)} < \infty$ denotes $L_{2}$ norm of $\bar{w} (t)$ .

Thus:

if $\bar{w} (t) = 0$ , then $v (T) - v (0) < - z^{T} (t) z (t) \leq 0$

if $\bar{w} (t) \neq 0$ , such that $‖ \bar{w} (t) ‖_{2}^{2} < ϖ^{2}$ then:

if $T \to \infty, \int_{0}^{\infty} z^{T} (t) z (t) < γ^{2} \int_{0}^{\infty} ({\bar{w}}^{T} (t) \bar{w} (t)) + v (0)$ , under the zero-initial condition, $\int_{0}^{\infty} z^{T} (t) z (t) < γ^{2} ‖ \bar{w} (t) ‖_{2}^{2}$

then, in the presence of disturbance, the system have H_∞ disturbance attenuation level $γ$ .

On the other hand, $v (T) \leq v (0) + γ^{2} \int_{0}^{T} {\bar{w}}^{T} (t) \bar{w} (t) < 1 + γ^{2} ϖ^{2}$ , i.e. $ε (P, ρ), ρ = 1 + γ^{2} ϖ^{2} \forall T > 0$ , is attraction domain.

The ellipsoid $ε (P, ρ)$ is inside $Ω$ if:

{\tilde{K}}_{η}^{T} (t) {(\frac{P}{ρ})}^{- 1} {\tilde{K}}_{η} \leq {\bar{U}}^{2}

(31)

The following theorem presents the tracking error for the PV system (equation (27)) under $H_{\infty}$ performance (equation (28)).

Theorem 1: If there exist common symmetric positive definite matrix Q , matrices Fj , Kj and scalars $γ > 0$ , solution of the following inequalities:

\begin{matrix} (\begin{matrix} ({\tilde{A}}_{l} Q + {\tilde{B}}_{i} {\tilde{F}}_{j}) + (*) & \bar{G} & Q {\tilde{C}}_{e}^{T} \\ * & - γ^{2} I & O \\ * & * & - I \end{matrix}) < 0 \\ \forall j \in I_{r}, \forall i \in I_{r}, \forall l \in I_{r_{ref}} \end{matrix}

(32)

\begin{matrix} (\begin{matrix} \frac{{\bar{U}}_{i}^{2}}{ρ} & k_{i}^{j} \\ * & Q \end{matrix}) < 0 \\ \forall j \in I_{r}, \forall i \in I_{r} \end{matrix}

(33)

where I is the identity matrix of appropriate dimension with ${\tilde{K}}_{η} = {\tilde{F}}_{η} Q^{- 1}$ .

Then the $H_{\infty}$ control performance in equation (28) is guaranteed for a prescribed attenuation level $γ^{2}$ .

Proof:

Let us consider the quadratic Lyapunov function candidate

v (t) = {\tilde{X}}^{T} (t) P \tilde{X} (t)

(34)

with $P = P^{T} > 0$ .

The derivative of the Lyapunov function is given by

\overset{\cdot}{v} (t) = {\overset{\cdot}{\tilde{X}}}^{T} (t) P \tilde{X} (t) + {\tilde{X}}^{T} (t) P \overset{\cdot}{\tilde{X}} (t)

(35)

\begin{matrix} \overset{\cdot}{v} (t) = {({\bar{A}}_{η} \tilde{X} + \bar{G} \bar{w})}^{T} P \tilde{X} + {\tilde{X}}^{T} P ({\bar{A}}_{η} \tilde{X} + \bar{G} \bar{w}) \\ \overset{\cdot}{v} (t) = {\tilde{X}}^{T} ({P \bar{A}}_{η} + (*)) \tilde{X} + {\tilde{X}}^{T} (P \bar{G}) \bar{w} + {\bar{w}}^{T} ({\bar{G}}^{T} P) \tilde{X} \end{matrix}

(36)

The inequality of equation (28) becomes

\begin{matrix} J = {\tilde{X}}^{T} ({P \bar{A}}_{η} + (*)) \tilde{X} + {\tilde{X}}^{T} (P \bar{G}) \bar{w} + {\bar{w}}^{T} ({\bar{G}}^{T} P) \tilde{X} \\ + z^{T} (t) z (t) - γ^{2} {\bar{w}}^{T} (t) \bar{w} (t) < 0 \end{matrix}

(37)

in matrix inequality form:

{(\begin{matrix} \tilde{X} \\ \bar{w} \end{matrix})}^{T} (\begin{matrix} {P \bar{A}}_{η} + (*) & P \tilde{G} \\ * & - γ^{2} I \end{matrix}) (\begin{matrix} \tilde{X} \\ \bar{w} \end{matrix}) + z^{T} (t) z (t) < 0

(38)

From equations (38) and (29) and by applying the Schur complement we obtain

Z = (\begin{matrix} {P \bar{A}}_{η} + (*) & P \bar{G} & {\tilde{C}}_{e}^{T} \\ * & - γ^{2} I & O \\ * & * & - I \end{matrix}) < 0

(39)

Pre- and post-multiplying equation (39) by $Γ = diag (Q, I, I)$ with $Q = P^{- 1}$ , we have

Z = (\begin{matrix} {\bar{A}}_{ijl}^{η} Q + (*) & \bar{G} & Q {\tilde{C}}_{e}^{T} \\ * & - γ^{2} I & O \\ * & * & - I \end{matrix}) < 0

(40)

From equation (27), we can obtain the following LMI:

(\begin{matrix} ({\tilde{A}}_{l} Q + {\tilde{B}}_{i} {\tilde{K}}_{j} Q) + (*) & \bar{G} & Q {\tilde{C}}_{e}^{T} \\ * & - γ^{2} I & O \\ * & * & - I \end{matrix}) < 0

(41)

Using matrices ${\tilde{F}}_{η} = {\tilde{K}}_{η} Q$ , the inequality (equation (41)) can be expressed in the form of equation (32).

We can obtain the LMI of equation (33) in the theorem by considering the LMI of equation (31) and using the same proof of the LMI of equation (32). End of Proof.

Simulation results

In order to check the effectiveness of the proposed method, we use a Lorentz solar PV module LC 120-15P, whose specifications are stated in Table 1. The parameters of the boost converter are shown in Table 2.

Table 1.

PV panel module LC120-12P.

Parameters	Symbol	Value
Series cells	N_s	36
Series parallel	N_p	1
Maximum power	P_pvOp	120 W
Maximum current	I_pvOp	7 A
Maximum voltage	V_pvOp	17.1 V
Open-circuit voltage	V_oc	7.7 V
Short-circuit current	I_ph	21.8 A

Table 2.

Boost converter parameters.

Parameters	Symbol	Value
Input capacitor	C_a	500 µF
Output capacitor	C_b	100 µF
Load resistance	R ₀	20 Ω
Inductor	L	10 mH
Resistance of self-inductance	R_L	0.01 Ω

The T-S Fuzzy controller gains are calculated by solving the LMIs (equations (29) and (30)) and the control parameters are given as follows:

$K_{1} = [2.928911485963169 - 9.3811613669238981.235277606074184],$

$K_{2} = [2.925833633295309 - 8.3737557519841511.234167434960502],$

$K_{3} = [3.903145593501021 - 9.3172476490933571.225551850241995],$

$K_{4} = [1.903145593501004 - 8.3172476490933111.225551850241991],$

$L_{1} = [0.016667711989449 - 0.0000018263319351.005697641010647],$

$L_{2} = [0.016653087263086 - 0.0000018261284341.005691402040381],$

$L_{3} = [1.016538275980667 - 0.0000018145847970.005643518806140],$

$L_{4} = [1.016538275980667 - 0.0000018145847970.005643518806140] .$

To test the effectiveness of the proposed MPPT fuzzy controller, the PV system responses are carried out by considering insolation and temperature variations. Moreover, a comparison with the following two control approaches is presented:

P&O and INC methods which are the most commonly used in practice because of their ease of implementation;

Conventional PI controller defined by: $U (t) = K_{p} . (V_{PVref} - V_{PV}) + K_{i} \int (V_{PVref} - V_{PV}) dt$ , where the control gains are $K_{p} = 0.035$ and $K_{i} = 0.7$ .

Three scenarios can appear.

Scenario 1: Constant temperature and shading phenomenon

In this case, the cell temperature is kept constant at the reference value T = 25°C and the insolation is varied with shading phenomenon as follows:

λ (W/m²)

200

1000

100

500

300

Scenario 2: Variable temperature and constant insolation

In this case, the insolation is kept constant at the reference value λ = 1000 W/m² and temperature is varied as follows:

T (°C)

Scenario 3: Variable temperature and shading phenomenon

In this case, the insolation and T are varied at the same time with shading phenomenon.

Figure 5 shows that the proposed T-S fuzzy controller presents a better MPPT performance and, especially, a very interesting settling time compared with the other methods and an output that follows accurately the reference signal with less transient-state oscillations. Table 3 gives a comparison between settling time of T-S controller and the other methods as well as the result obtained by Allouche et al. (2018). We observe that the settling time is not exceeding 0.017 (s) whereas PI, PO controller and Allouche et al. (2018) methods present a settling time exceeding 0.1 (s).

Table 3.

Settling time controller.

Settling time (s)
$λ$ (W/m²)	T (°C)	T-S fuzzy controller	PI controller	PO controller	INC controller	Allouche et al. (2018)
300	20	0.01	0.19	0.17	0.1	0.1
500	25	0.01	0.25	0.24	0.14	0.1
700	30	0.011	0.28	0.28	0.18	0.1
900	35	0.017	0.29	0.31	0.21	0.1
1000	40	0.017	0.3	0.32	0.22	0.1
1000	25	0.017	0.3	0.33	0.23	0.1

Figure 6 shows the PV power response under changing climatic conditions (different scenarios). We can see that the maximum power of T-S controller is achieved with a very low-power tracking error without including ripple behaviour. This proves that maximum power tracking is achieved for shading phenomenon (scenarios 1 and 3) and for all temperature variations (scenarios 2 and 3), and hence, the PV system is operating on its optimum performance. However, the PI controller exhibits sensitivity to an abrupt insolation and temperature changes resulting a ripple during the transient period which leads to an inexact MPP voltage tracking. The P&O controller is characterized by a slower response time and it presents a performance which is not suitable for a steady state because it forces the output power to oscillate around the MPP mainly at high insolation levels. The critical factor of the INC algorithm is the perturbation rate to have minimum oscillation around the operating point.

Consequently, the proposed method is more efficient than the classical methods in terms of convergence speed and tracking accuracy. On the other hand, the dynamic behaviour of the power response of the conventional PI controller is slower and present oscillations in transient period. The P&O and the INC methods exhibit a poor dynamic response with large oscillations around the MPP.

Figure 7 shows the PV power response, the maximum power is reached in a reduced time comparing it with other methods. It does not show ripple during the transient phase and it illustrated a low-power tracking error. This confirms that almost full available power is extracted from the PV generator, and hence, the PV system is operating on its optimum performance. The dynamic behaviour for the conventional PI controller of the power response is slower as it reaches the MPP within 0.3 s mainly at nominal insolation values (1000 W/m² at 25°C) and it is characterized by a slight fluctuation during the transient period. P&O and INC algorithms present oscillations around the optimal value.

The major disadvantage is a bad behaviour for a sudden change insolation. Therefore, the advantage of the proposed method-based reference model is improving the efficiency of the system by giving a higher power.

In Table 4, we compare the reference power P_pvref computed by the T-S reference model for each temperature and insolation pair to the PV module power, and generated successively by the use of the T–S fuzzy controller, PI controller, P&O and INC methods and the result obtained by Allouche et al. (2018).

Table 4.

Comparison of power extracted from PV module.

$λ$ (W/m²)	T (°C)	Pref (W)	P-TS (W)	(Err)_TS (%)	P-PI (W)	(Err)_PI (%)	P-Vco (W)	(Err)_Vco (%)	P-INC (W)	(Err)_INC (%)	P-PO (W)	(Err)_PO(%)	(Err)_TS (Allouche et al., 2018) (%)
300	20	33.01	32.96	0.05	32.96	0.05	34.1	1.09	35.16	2.15	32.42	0.59	1.08
500	25	56.2	56.08	0.12	56.06	0.14	54.09	2.11	53.39	2.81	56.54	0.34	0.1
700	30	79.07	78.84	0.23	78.81	0.26	80.23	1.16	78.64	0.43	79.65	0.58	0.85
900	35	101.4	101	0.4	100.9	0.5	99.23	2.17	103.2	1.8	102.5	1.1	1.83
1000	40	110.6	110.1	0.5	110.1	0.5	114.6	4	107.4	3.2	108.5	2.1	2.67
1000	25	120.5	120	0.5	119.8	0.7	112.4	8.1	109.4	11.1	119.1	1.4	-

Figure 8 shows the controller effect on PV array that operates very close to its maximum power trajectory, which leads to an important transfer of the available solar power and an improved PV system performance. For the PI controller use, there is a slight deviation from the MPP trajectory with the decrease in the insolation. For the P&O and INC algorithm, its strongly depends on the initial conditions and it presents oscillations around the optimal value. The major disadvantage of this algorithm is its bad behaviour following a sudden change in illumination.

Conclusion

This paper presents an H_∞ fuzzy PI state feedback controller for maximum power point (MPP) tracking of PV systems subject to actuator saturation in order to minimize tracking error and disturbance effect. Based on TS fuzzy representation, H∞ stabilization conditions of the closed-loop system are formulated and solved in terms of linear matrix inequalities (LMIs) terms. The obtained simulation results show that the proposed controller is able to track the MPP with fast convergence even under changing climatic conditions. Future works will focus on experimental validation and the consideration of input DC-DC constraint.

Footnotes

Authors’ Note

K Houda is also affiliated with LAJ, University of Jijel, Algeria.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

K Houda

References

Abdessalem

Massoud

Ahmed

, et al. (2011) High performance adaptive perturb and observe MPPT technique for photovoltaic based microgrids. IEEE Transactions on Power Electronics 26(4): 1010–1021.

Ajaamoum

Kourchi

Bouachrine

, et al. (2015) Comparison of Takagi–Sugeno fuzzy controller and the command “P & O” for extracting the maximum power from a photovoltaic system. International Journal of Innovation and Applied Studies 10(1): 192-206.

Allouche

Dahech

Chaabane

(2018) Multiobjective maximum power tracking control of photovoltaic systems: T-S fuzzy model-based approach. Soft Computing 22(7): 2121-2132.

Azri

Rahim

Elias

MFM

(2014) Transformer less DC/AC converter for grid-connected PV power generation system. Arabian Journal for Science and Engineering 39(11): 7945–7956.

Brunton

Rowley

Kulkarni

, et al. (2010) Maximum power point tracking for photovoltaic optimization using ripple-based extremum seeking control. IEEE Transactions on Power Electronics 25(10): 2531–2540.

Cao

Lin

(2003a) Stability analysis of discrete-time systems with actuator saturation by a saturation-dependent Lyapunov function. Automatica 39(7): 1235–1241.

Cao

Lin

(2003b) Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation. IEEE Transactions on Fuzzy Systems 11(1): 57-67.

Chiu

(2010) T-S fuzzy maximum power tracking control of solar power generation systems. IEEE Transactions on Energy Conversion 25(4): 1123–1132.

Chiu

Ouyang

(2011) Robust maximum power tracking control of uncertain photovoltaic systems: a unified T-S fuzzy model-based approach. IEEE Transactions on Control Systems Technology 19(6): 1516–1526.

10.

Cossutta

Aguirre

Cao

, et al. (2015) Single stage fuel cell to grid interface with multilevel current-source inverters. IEEE Transactions on Industrial Electronics 62(8): 5256–5264.

11.

Dahmane

Bosche

El-Hajjaji

, et al. (2013) T-S implementation of an MPPT algorithm for photovoltaic conversion system using poles placement and H_∞ performances. In: Proceedings of the 3rd international conference on systems and control, Algiers, 2013, 29–31 October 2013, pp. 1116–1121.

12.

DaSilva

Fischman

Tarbouriech

, et al. (1997) Synthesis of state feedback for linear systems subject to control saturation by an LMI-based approach. IFAC Proceedings Volumes 30(16): 207-212.

13.

De Brito

MAG

Galotto

Sampaio

, et al. (2013) Evaluation of the main MPPT techniques of photovoltaic applications. IEEE Transactions on Industrial Electronics 60(3): 1156–1167

14.

Estrada-Manzo

Lendek

Guerra

(2019) An alternative LMI static output feedback control design for discrete-time nonlinear systems represented by Takagi-Sugeno models. ISA Transactions 84: 104-110.

15.

Hafedh

Fernando

Mansour

(2012) Maximum power point tracking for photovoltaic panel based on T-S fuzzy systems. International Journal of Computer Applications 44(22): 50-58.

16.

Han

(2007) T-S fuzzy controller in consideration of input constraint. In: Proceedings of the 2nd international conference on innovative computing, information and control, Kumamoto, Japan, 5–7 September 2007.

17.

Henrion

(1999) Stabilité des systèmes linéaires incertains à commande contrainte. PhD Thesis, Institut National des Sciences Appliquées de Toulouse, France.

18.

Khabou

Souissi

Aitouche

(2020) MPPT implementation on boost converter by using T-S fuzzy method. Mathematics and Computers in Simulation 167: 119-134.

19.

Lien

C-H

Vaidyanathan

K-W

, et al. (2019) Robust mixed performance for uncertain Takagi-Sugeno fuzzy time-delay systems with linear fractional perturbations. International Journal of Modelling, Identification and Control 31(2): 193–203.

20.

Liu

Duan

Liu

, et al. (2008) A variable step size INC MPPT method for PV systems. IEEE Transactions on Industrial Electronics 55(7): 2622–2628.

21.

Lyden

Haque

(2015) Maximum power point tracking techniques for photovoltaic systems: a comprehensive review and comparative analysis. Renewable and Sustainable Energy Reviews 52: 1504–1518.

22.

Makhloufi

(2014) Contribution à l’optimisation des installations photovoltaïques par des commandes intelligentes. Thèse de doctorat en sciences, Université de Batna [Contribution to the optimization of photovoltaic installations by intelligent controls. Doctoral thesis in science, University of Batna].

23.

Masoum

MAS

Dehbonei

Fuchs

(2002) Theoretical and experimental analysis of photovoltaic systems with voltage and current-based maximum power-point tracking. IEEE Transactions on Energy Conversion 17(4): 514–522.

24.

Mastromauro

Liserre

Kerekes

, et al. (2009) A single-phase voltage-controlled grid-connected photovoltaic system with power quality conditioner functionality. IEEE Transactions on Industrial Electronics 56(11): 4436–4444.

25.

Meghni

Dib

Azar

, et al. (2018) Effective supervisory controller to extend optimal energy management in hybrid wind turbine under energy and reliability constraints. International Journal of Dynamics and Control 6: 360–383.

26.

Mohammed

ASM

Hooman

Ewald

(2002) Theoretical and experimental analyses of photovoltaic systems with voltage-and current-based maximum power-point tracking. IEEE Transactions on Energy Conversion 17(4): 514–522.

27.

Nabulsi

Dhaouadi

(2012) Efficiency optimization of a DSP-based standalone PV system using fuzzy logic and dual-MPPT control. IEEE Transactions on Industrial Informatics 8(3): 573–584.

28.

Nasri

Saifia

Chadli

, et al. (2019) H_∞ static output feedback control for electrical power steering subject to actuator saturation via fuzzy Lyapunov functions. Transactions of the Institute of Measurement and Control 41(12): 3340–3351.

29.

Ohtake

Tanaka

Wang

(2003) Fuzzy modeling via sector nonlinearity concept. Integrated Computer-Aided Engineering 10(4): 333–341.

30.

Priyabrata

Priyabrat

(2019) Analysis, design and implementation of analog circuitry-based maximum power point tracking for photovoltaic boost DC/DC converter. Transactions of the Institute of Measurement and Control 41(3): 668–686.

31.

Rahim

AHMA

Khan

(2014) A swarm-based adaptive neural network SMES control for a permanent magnet wind generator. Arabian Journal for Science and Engineering 39(11): 7957–7965.

32.

Saifia

Chadli

Labiod

(2010a) Static output feedback stabilization of multiple models subject to actuators saturation. In: 12th IFAC LSS symposium, Villeneuve d’Ascq, France, 11–14 July 2010.

33.

Saifia

Chadli

Labiod

(2010b) H_∞ stabilization of quarter-car suspension system subject to actuator saturation. In: ICRE’07, Bejaia, Algeria, 2–3 November 2010.

34.

Saifia

Chadli

Labiod

(2011) H_∞ control of multiple model subject to actuator saturation: application to quarter-car suspension system. Analog Integrated Circuits and Signal Journal 69: 81–90.

35.

Saifia

Chadli

Labiod

, et al. (2012a) Robust H_∞ static output feedback stabilization of T-S fuzzy systems subject to actuator saturation. International Journal of Control, Automation, and Systems 10(3): 613–622.

36.

Saifia

Chadli

Labiod

(2012b) Robust stabilization of PMSM system subject to actuator saturation. In: ICIPEE, Tebessa, Algeria, 14–16 April 2012.

37.

Saifia

Chadli

Labiod

(2012c) H_∞ multiple model control of suspension system with actuators saturation. In: Handbook of Vehicle Suspension Control Systems IET, pp. 195–246.

38.

Saifia

Chadli

Labiod

, et al. (2012d) Fuzzy control of DC-DC converters with input constraint. Mathematical Problems in Engineering 2012: 973082.

39.

Saifia

Chadli

Labiod

, et al. (2019) Robust H_∞ static output-feedback control for discrete-time fuzzy systems with actuator saturation via fuzzy Lyapunov functions. Asian Journal of Control. Epub ahead of print. DOI: 10.1002/asjc.1987.

40.

Saifia

Chadli

Labiod

, et al. (2020) Robust H_∞ static output-feedback control for discrete-time fuzzy systems with actuator saturation via fuzzy Lyapunov functions. Asian Journal of Control 22(2): 611–623.

41.

Salah

Ouali

(2011) Comparison of fuzzy logic and neural network in maximum power point tracker for PV systems. Electric Power Systems Research 81(1): 43–50.

42.

Sera

Mathe

Kerekes

, et al. (2013) On the perturb-and-observe and incremental conductance MPPT methods for PV systems. IEEE Journal of Photovoltaics 3(3): 1070–1078.

43.

Singh

Subhash

(2016) Optimal sizing of grid integrated hybrid PV-biomass energy system using artificial bee colony algorithm. IET Renewable Power Generation 10(5): 642–650.

44.

Subudhi

Pradhan

(2012) A comparative study on maximum power point tracking techniques for photovoltaic power systems. IEEE Transactions on Sustainable Energy 4(1): 89–98.

45.

Takagi

Sugeno

(1985) Fuzzy identification of systems and its applications to modelling and control. IEEE Transactions on Systems, Man, and Cybernetics 15(1): 116-132.

46.

Vahid

Sheikholeslam

Mohammad

RJM

(2018) Maximum power point tracking with constraint feedback linearization controller and modified incremental conductance algorithm. Transactions of the Institute of Measurement and Control 40(7): 2322-2331.

47.

Vincheh

Kargar

Markadeh

(2014) A hybrid control method for maximum power point tracking (MPPT) in photovoltaic systems. Arabian Journal for Science and Engineering 39(6): 4715–4725.

48.

Wang

Tanaka

Griffin

(1996) An approach to fuzzy control of nonlinear systems: stability and design issues. IEEE Transactions on Fuzzy Systems 4(1): 14–23.

49.

Xin

Shuang

Xian

, et al. (2019) Further studies on robust H_∞ control for a class of Takagi–Sugeno fuzzy time-delay systems with application to continuously stirred tank reactor problems. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 233(2): 103-117.

H ∞ fuzzy proportional integral state feedback controller of photovoltaic systems under asymmetric actuator constraints

Abstract

Keywords

Introduction

Fuzzy modelling of PV generator systems

Photovoltaic generator model

DC-DC boost converter

T-S fuzzy model of the PV system

Robust T-S fuzzy MPPT control design under asymmetric input constraint

Reference model

Location of MPP (VpvOp, IpvOp)

H∞ tracking control design under asymmetric input constraint

Simulation results

Conclusion

Footnotes

Authors’ Note

Declaration of Conflicting Interests

Funding

ORCID iD

References

Location of MPP (V_pvOp, I_pvOp)

H_∞ tracking control design under asymmetric input constraint