Decentralized T-S Fuzzy Control for Solar PV Powered Water Pumping System Driving by Induction Motor

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Figure 1. Photovoltaic pumping system

The above Figure 1 presents the structure of the proposed PV pumping system which is composed of the first subsystem: PV panel connected to a boost converter; a T-S fuzzy controller is developed to ensure the MPPT, and a second subsystem: an inverter connected to the unit motor-pump (centrifugal pump); equally a T-S fuzzy controller is dedicated for trajectory tracking under variable solar irradiation and ambient temperature.  

2.1 PV conversion system

2.1.1 PV module

In the case of high-power applications and in order to reach the desired voltage and current levels, solar modules are electrically wired together to obtain a continuous electric source known as a photovoltaic generator (PVG). In this work, the PVG used is composed of 10 LC120P solar panels connected in series. Its parameters are presented in the Table 1.

Table 1. LC120-12P module

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Figure 2. Solar cell equivalent circuit

According to the above Figure 2, the photovoltaic current is given by the following expression:

$I_{p v}=I_{p h}-I_{0}\left(\exp \left(\frac{I_{p v}+R_{s} I_{p v}}{n V_{T}}\right)-1\right)-\left(\frac{V_{p v}+R_{s} I_{p v}}{R_{s h}}\right)$   (1)

where, VT=KT/q; K is the Boltzmann constant, T is the cell temperature, q is the elementary charge, RS and Rsh are series and parallel resistances, respectively, I_ph is the generated photo current depending on the insolation and temperature changes and I_0 represents the diode’s saturation current. The following Figure 3 indicates the Power–Voltage (P-V) characteristic of the PV module. The curve shows the location of the MPP of the PV panel under various values of solar irradiation and constant cell temperature (25℃).

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Figure 3. Characteristics of the PV module for various values of insolation

2.1.2 DC/DC boost converter modeling

Figure 4 shows the basic circuit topology of the DC-DC boost converter. Its dynamic can be expressed with a linear differential equation that correspond to two phases: the store phase of electrical energy in the inductor (the switch is ON) and the transfer phase of energy (the switch is OFF). This leads to an average model as follows:

$\begin{aligned} \dot{x_{1}}(t)=&\left[A_{11} x_{1}(t)+E_{1} w_{1}(t)\right] u(t) \\ &+\left[A_{12} x(t)+E_{1} w_{1}(t)\right](1-u(t)) \end{aligned}$    (2)

As well,

$\dot{x_{1}}(t)=A_{12} x_{1}(t)+\left(A_{11}-A_{12}\right) x_{1}(t) u(t)$$+E_{1} w_{1}(t)$      (3)

$\dot{x_{1}}(t)=A_{12} x_{1}(t)+B_{1}\left(x_{1}(t)\right) u(t)+E_{1} w_{1}(t)$    (4)

where, $u(t) \in[0,1]$

$A_{11}=\left(\begin{array}{ccc}0 & -\frac{1}{c_{1}} & 0 \\ \frac{1}{L} & -\frac{R_{L}}{L} & 0 \\ 0 & 0 & -\frac{1}{R C_{2}}\end{array}\right)$; $A_{12}=\left(\begin{array}{ccc}0 & -\frac{1}{c_{1}} & 0 \\ \frac{1}{L} & -\frac{R_{L}}{L} & -\frac{1}{L} \\ 0 & \frac{1}{C_{2}} & -\frac{1}{R C_{2}}\end{array}\right)$;

$E_{1}=\left(\begin{array}{c}\frac{1}{C_{1}} \\ 0 \\ 0\end{array}\right)$    $x_{1}(t)=\left(\begin{array}{c}V_{p v}(t) \\ I_{L}(t) \\ V_{c 2}(t)\end{array}\right)$     $w_{1}(t)=I_{p v}(t)$

where, $B_{1}\left(x_{1}(t)\right)=\left(\begin{array}{c}0 \\ \frac{V_{c 2}(t)}{L} \\ -\frac{I_{L}(t)}{C_{2}}\end{array}\right)$ and u(t) defines the duty cycle.

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Figure 4. DC/DC boost converter

2.1.3 MPPT searching algorithm

The MPPT algorithm adapted in our work, is based on the instant computation of the partial derivative of PV power with respect to the PV current. The MPP searching bloc gets the insolation and the temperature as inputs and provides the optimal values of PV voltage and current. At a maximum power point, the following condition is verified [11]:

$\frac{d}{d I_{p v}}\left(V_{p v} I_{p v}\right)=0$   (5)

Therefore,

$V_{p v}+I_{p v} \frac{d V_{p v}}{d I_{p v}}=0$    (6)

For maximum values of PV current and voltage, the value of PV power is maximum. By substituting V_pvopt and $\frac{d V_{p v}}{d I _{p v}}$ by their values in in the above equation, we obtain the following equation:

$\frac{2 R_{s}}{n V_{T}} I_{\text {pvopt}}-\left(I_{p h}-I_{\text {pvopt}}+I_{0}\right) Ln \left(\frac{I_{p h}-I_{\text {pvopt}}+I_{0}}{I_{0}}\right)=0$    (7)

The previous equation is solved for I_pvopt and indicates that this latter is proportionally dependent on the cell photocurrent I_ph: 

$I_{ {pvopt}}=0.9091 I_{p h}$   (8)

2.1.4 T-S fuzzy model for boost converter

The Takagi-Sugeno models are considered as universal approximators that ensure the perfect description of the dynamic behavior of nonlinear systems. The premise variables  $z_{1 k}(t)=x_{1 k}(t) \in\left[z_{1 k, \min }, z_{1 k, \min }\right], k=1,2$ of the PV system are chosen as follow:

$\left\{\begin{array}{c}z_{11}(t)=V_{c 2}(t) \\ z_{12}(t)=I_{L}(t)\end{array}\right.$

Thus, the membership functions are defined by the following expressions:

$F_{1 k, \min }(z(t))=\frac{z_{1 k}(t)-z_{1 k, \min }}{z_{1 k, \max }-z_{1 k, \min }}$

$F_{1 k, \max }(z(t))=1-F_{1 k, \min }(z(t))$

The weighting functions of the derived T-S model are given by:

$\begin{aligned} h_{1}(z(t)) &=F_{11, \min }(z(t)) * F_{12, \min }(z(t)) \\ h_{2}(z(t)) &=F_{11, \min }(z(t)) * F_{12, \max }(z(t)) \\ h_{3}(z(t)) &=F_{11, \max }(z(t)) * F_{12, \min }(z(t)) \\ h_{4}(z(t)) &=F_{11, \max }(z(t)) * F_{12, \max }(z(t)) \end{aligned}$

Eventually, the overall output of the fuzzy rule-based system is given by:

$\dot{x_{1}}(t)=\sum_{i=1}^{4} h_{i}(z(t))\left(A_{12} x_{1}(t)+B_{1 i} u_{1}(t)+E_{1} w_{1}(t)\right)$    (9)

$B_{11}=\left(\begin{array}{c}0 \\ \frac{V_{c 2 \min }(t)}{L} \\ \frac{I_{L \min }(t)}{C_{2}}\end{array}\right)$;  $B_{12}=\left(\begin{array}{c}0 \\ \frac{V_{C 2 \min }(t)}{L} \\ \frac{I_{L \max }(t)}{C_{2}}\end{array}\right)$

$B_{13}=\left(\begin{array}{c}0 \\ \frac{V_{C 2 \max }(t)}{L} \\ \frac{I_{{Lmin}}(t)}{C_{2}}\end{array}\right)$;  $B_{14}=\left(\begin{array}{c}0 \\ \frac{V_{C 2 \max }(t)}{L} \\ \frac{I_{{Lmax}}(t)}{C_{2}}\end{array}\right)$

2.1.5 T-S fuzzy reference model

The goal to reach from this work is to elaborate a T-S fuzzy controller capable of ensuring the tracking of an optimal reference model which will produce the appropriate reference state to track. The equation of the MPP reference model in the state form can be written as follow [11]:

$\dot{x}_{1 r}(t)=A_{1 r} x_{1 r}(t)+r_{1}(t)$  (10)

where,

$A_{1 r}=\left(\begin{array}{ccc}0 & -\frac{1}{C_{1}} & 0 \\ \frac{1}{L} & -\frac{R_{L}}{L} & -\frac{1}{L}\left(1-u_{o p t}\right) \\ 0 & \frac{1}{C_{2}}\left(1-u_{o p t}\right) & -\frac{1}{R C_{2}}\end{array}\right)$

$u_{o p t}=\sqrt{\frac{V_{p v o p t}}{R I_{p v o p t}}}$

The reference model (10) is also nonlinear via the premise variable $z_{1 r}=\left(1-u_{o p t}\right)$. Thereby, the membership functions are presented by the following expressions:

$N_{1 \min }\left(z_{1 r}(t)\right)=\frac{z_{1 r}(t)-z_{1 r, \min }}{z_{1 r, \max }-z_{1 r, \min }}$

The matrices of the reference model are defined as:

$A_{11 r}=\left(\begin{array}{ccc}0 & -\frac{1}{C_{1}} & 0 \\ \frac{1}{L} & -\frac{R_{L}}{L} & -\frac{1}{L} z_{r, \min } \\ 0 & \frac{1}{C_{2}} z_{r, \min } & -\frac{1}{R C_{2}}\end{array}\right)$

$A_{12 r}=\left(\begin{array}{ccc}0 & -\frac{1}{C_{1}} & 0 \\ \frac{1}{L} & -\frac{R_{L}}{L} & -\frac{1}{L} z_{r, \max } \\ 0 & \frac{1}{C_{2}} z_{r, \max } & -\frac{1}{R C_{2}}\end{array}\right)$

As so, the global T-S fuzzy reference model is:

$\dot{x}_{1 r}(t)=\sum_{k=1}^{2} h_{k}\left(z_{1 r}(t)\right)\left(A_{1 k r} x_{1 r}(t)+r_{1}(t)\right)$    (11)

2.1.6 T-S fuzzy controller

This section is dedicated to develop the control laws which are also based on the T-S fuzzy modeling. Here, we adopted the same principle of the ordinary PDC controller but in addition, in the control law, we take into account the term of the tracking error: $e_{1 r}(t)=\left(x_{1}(t)-x_{1 r}(t)\right)$. As mentioned before, the objective is that, in spite of the change in the insolation values, the system needs to function in the optimum operating conditions which are characterized by a tracking error converging to zero. Thus, the trajectory tracking issue is formulated as a fuzzy state feedback control given by:

$u_{1}(t)=\sum_{j=1}^{4} h_{j}(z(t)) K_{1 j}\left(e_{1 r}(t)\right)$    (12)

where, K_1j are the controller gains to calculate later.

By replacing each of the reference state (11), the system real state (9) and the control law (12) by their expressions, we obtain the error dynamics as:

$e_{1 r}^{\cdot}(t)$$=\sum_{i=1}^{4} \sum_{j=1}^{4} \sum_{k=1}^{2} h_{i}(z(t)) h_{j}(z(t)) h_{k}\left(z_{r}(t)\right)$

$\times\left[\begin{array}{c}\left(A_{12}+B_{1 i} K_{1 j}\right) e_{1 r}(t)+ \left(A_{12}-A_{1 k r}\right) x_{r 1}(t)+E_{1} w_{1}(t)-r_{1}(t)\end{array}\right]$    (13)

Thus, by injection the above expression in the overall T-S fuzzy model (9), the following augmented state-space form is obtained:

$\dot{\bar{{x}_{1}}}(t)=\sum_{i=1}^{4} \sum_{i=1}^{4} \sum_{k=1}^{2} h_{i}(z(t)) h_{j}(z(t)) h_{k}\left(z_{1 r}(t)\right)\times\left[\bar{A}_{i j k} \overline{x_{1}}(t)+\overline{E_{1}} \overline{w_{1}}(t)\right]$  (14)

where, 

$\begin{aligned} \overline{x_{1}}(t)=&\left[\begin{array}{l}e_{r 1}(t) \\ x_{1 r}(t)\end{array}\right] \quad \overline{w_{1}}(t)=\left[\begin{array}{c}w_{1}(t) \\ r_{1}(t)\end{array}\right] \quad \overline{E_{1}}=\left[\begin{array}{cc}E_{1} & -I \\ 0 & I\end{array}\right] \\ \bar{A}_{i j k} &=\left[\begin{array}{cc}A_{12}+B_{1 i} K_{1 j} & A_{12}-A_{1kr} \\ 0 & A_{1 k r}\end{array}\right] \end{aligned}$

r₁(t) is the external input depending on the insolation and the temperature and W₁(t) designates the external disturbance.

The H∞ performance is added to the closed-loop control system in the purpose of attenuating the external disturbance’s effect; it is expressed as follow:

$\int_{0}^{t f} \overline{x_{1}}^{T}(t) \overline{Q_{1}} \overline{x_{1}}(t) d t \leq \rho_{1}^{2} \int_{0}^{t f} \overline{w_{1}}^{T}(t) \overline{w_{1}}(t) d t$  (15)

where, $\overline{Q_{1}}=\left[\begin{array}{ll}Q_{1} & 0 \\ 0 & 0\end{array}\right]$ is a positive definite weighting matrix and ρ₁ is a prescribed attenuation level against the external disturbance effect.

2.2 Induction Motor modeling and control

To elaborate the electromagnetic dynamics of the IM, we are looking to determine the biphasic model of the machine in the synchronously d-q reference frame, in order to facilitate its study. In this context, these hypotheses are adopted:

    The magnetic circuit is linear.

    The permeability of the magnetic circuit is assumed to be infinite and iron losses are not taken into account.

    The mechanical losses are neglected.

The air gap thickness is constant while the notch effect is neglected.

2.2.1 Mathematical model in d-q coordinates

The dynamic model of the IM in the synchronous biphase d-q reference frame can be formulated as follow [17]:

$\dot{x_{2}}(t)=f_{2}\left(x_{2}(t)\right)+g_{2}\left(x_{2}(t)\right) u_{2}(t)+w_{2}(t)$  (16)

$g_{2}\left(x_{2}(t)\right)=\left[\begin{array}{ccccc}\frac{1}{\sigma L_{s}} & 0 & 0 & 0 & 0 \\ 0 & \frac{1}{\sigma L_{s}} & 0 & 0 & 0\end{array}\right]^{T}$; $u_{2}(t)=\left[\begin{array}{ll}u_{s d} & u_{s q}\end{array}\right]^{T}$

$x_{2}(t)=\left[\begin{array}{lllll}i_{s d} & i_{s q} & \psi_{r d} & \psi_{r q} & w_{m}\end{array}\right]^{T}$

$w_{2}(t)=\left[\begin{array}{lllll}0 & 0 & 0 & 0 & -\frac{C_{r}}{J}\end{array}\right]^{T}$

$f_{2}\left(x_{2}(t)\right)=\left[\begin{array}{c}-\alpha i_{s d}+w_{s} i_{s q}+\frac{K_{s}}{T_{r}} \psi_{r d}+K_{s} n_{p} w_{m} \psi_{r q} \\ -w_{s} i_{s d}-\alpha i_{s q}-K_{s} n_{p} w_{m} \psi_{r d}+\frac{K_{s}}{T_{r}} \psi_{r q} \\ \frac{M}{T_{r}} i_{s d}-\frac{1}{T_{r}} \psi_{r d}+\left(w_{s}-n_{p} w_{m}\right) \psi_{r q} \\ \frac{M}{T_{r}} i_{s q}-\left(w_{s}-n_{p} w_{m}\right) \psi_{r d}-\frac{1}{T_{r}} \psi_{r q} \\ \frac{n_{p} M}{J L_{r}}\left(\psi_{r d} i_{s q}-\psi_{r q} i_{s d}\right)-\frac{f}{J} w_{m}\end{array}\right]$

$\alpha=\left(\frac{1}{\sigma T_{s}}+\frac{1-\sigma}{\sigma T_{r}}\right) ; K_{S}=\frac{M}{\sigma L_{s} L_{r}} ; T_{r}=\frac{L_{r}}{R_{r}} ; T_{s}=\frac{L_{s}}{R_{s}}; \sigma=1-\frac{M^{2}}{L_{s} L_{r}}$

where, w_s is the electrical speed of stator, wm is the rotor speed, (u_sd;u_sq) are the direct and quadratic voltages of the stator, (i_sd;i_sq) are the stator currents and (ψ_rd;ψ_rq) are the flux of the rotor, C_r is the load torque, J is the moment of inertia, (R_s;R_r) are the rotor and stator resistances, (L_s;L_r) are the rotor and stator inductances, M is the mutual-inductance, f is the friction coefficient and n_p is the number of pole pairs.

2.2.2 Open loop control strategy

In this section we adopted the principle of rotor flow orientation for the open loop control, it ensures a natural decoupling between the stator current i_sd and the rotor flux ψ_rd and between the stator current i_sq and the electromagnetic torque C_em. By replacing the state variables $x_{2}(t)=\left[\begin{array}{lllll}i_{s d c} & i_{s q c} & \psi_{r d c} & \psi_{r q} & w_{m c}\end{array}\right]^{T}$ of the machine $\begin{array}{llll}\text { by } & \text { its } & \text { reference } & \text { corresponding } & \text { ones } & x_{2 r}(t)=\end{array}$ $\left[\begin{array}{lllll}i_{s d c} & i_{s q c} & \psi_{r d c} & 0 & w_{m c}\end{array}\right]^{T}$ in $(16),$ we obtain the following model:

$\left\{\begin{array}{c}\frac{d}{d t} i_{s d c}=-\alpha i_{s d c}+w_{s} i_{s q c}+\frac{K_{s}}{T_{r}} \psi_{r d c}+\frac{1}{\sigma L_{s}} u_{s d c} \\ \frac{d}{d t} i_{s q c}=-w_{s} i_{s d c}-\alpha i_{s q c}-K_{s} n_{p} w_{m c} \psi_{r d c}+\frac{1}{\sigma L_{s}} u_{s q c} \\ \frac{d}{d t} \psi_{r d c}=\frac{M}{T_{r}} i_{s d c}-\frac{1}{T_{r}} \psi_{r d c} \\ 0=\frac{M}{T_{r}} i_{s q c}-\left(w_{s}-n_{p} w_{m c}\right) \psi_{r d c} \\ \frac{d}{d t} w_{m c}=\frac{n_{p} M}{J L_{r}}\left(\psi_{r d c} i_{s q c}\right)-\frac{f}{J} w_{m c}-\frac{c_{r}}{J}\end{array}\right.$    (17)

From the above system, we extract the reference stator current, the reference electrical speed and the open-loop control law:

$\left\{\begin{array}{c}i_{s d c}=\frac{\psi_{r d c}}{M}+\frac{T_{r}}{M} \frac{d}{d t} \psi_{r d c} \\ i_{s q c}=\frac{J L_{r}}{n_{p} M \psi_{r d c}}\left(\frac{C_{r}}{J}+\frac{f}{J} w_{m c}+\frac{d}{d t} w_{m c}\right)\end{array}\right.$   (18)

$w_{s c}=n_{p} w_{m c}+\frac{M}{T_{r} \psi_{r d c}} i_{s q c}$    (19)

$\left\{\begin{array}{c}u_{s d c}=\sigma L_{s}\left(\frac{d}{d t} i_{s d c}+\alpha i_{s d c}-w_{s c} i_{s q c}-\frac{K_{s}}{T_{r}} \psi_{r d c}\right) \\ u_{s q c}=\sigma L_{s}\left(\frac{d}{d t} i_{s q c}+\alpha i_{s q c}+w_{s c} i_{s d c}+K_{s} n_{p} w_{m c} \psi_{r d c}\right)\end{array}\right.$  (20)

2.2.3 T-S fuzzy model of the Induction Motor

When the field-oriented strategy is implemented, the dynamics of the IM is similar to the separately excited DC motor and the electrical stator speed is expressed in the synchronously rotating frame as:

$w_{s}=n_{p} w_{m}+\frac{M}{T_{r} \psi_{r d c}} i_{s q}$     (21)

Thus, the nonlinear model of the IM may be given as the state space form:

$\left\{\begin{array}{c}\dot{x_{2}}(t)=A_{2} x_{2}(t)+B_{2} u_{2}(t)+w_{2}(t) \\ y_{2}(t)=C_{2} x_{2}(t)\end{array}\right.$    (22)

$A_{2}\left(x_{2}(t)\right)=\left[\begin{array}{ccccc}-\alpha & w_{s} & \frac{K_{s}}{T_{r}} & K_{s} n_{p} w_{m} & 0 \\ -w_{s} & -\alpha & -K_{s} n_{p} w_{m} & \frac{K_{s}}{T_{r}} & 0 \\ \frac{M}{T_{r}} & 0 & -\frac{1}{T_{r}} & \frac{M}{T_{r} \psi_{r d c}} i_{s q} & 0 \\ 0 & \frac{M}{T_{r}} & -\frac{M}{T_{r} \psi_{r d c}} i_{s q} & -\frac{1}{T_{r}} & 0 \\ 0 & 0 & \frac{n_{p} M}{J_{r}} i_{s q} & -\frac{n_{p} M}{J L_{r}} i_{s d} & -\frac{f}{J}\end{array}\right]$

$B_{2}=\left[\begin{array}{ccccc}\frac{1}{\sigma L_{s}} & 0 & 0 & 0 & 0 \\ 0 & \frac{1}{\sigma L_{s}} & 0 & 0 & 0\end{array}\right]^{T}$

$C_{2}=\left[\begin{array}{lllll}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1\end{array}\right] ; w_{2}(t)=\left[\begin{array}{lllll}0 & 0 & 0 & 0 & -\frac{C_{r}}{J}\end{array}\right]^{T}$

$A_{2}\left(x_{2}(t)\right)$ includes nonlinear terms $z_{2 k}(t)=x_{2 k}(t) \in$ $\left[z_{2 k m i n}, z_{2 k m a x}\right]$ with $k=1,2,3,$ as follow:

$\left\{\begin{array}{l}z_{21}(t)=i_{s d}(t) \\ z_{22}(t)=i_{s q}(t) \\ z_{23}(t)=w_{m}(t)\end{array}\right.$   (23)

Thus, the membership functions corresponding to the premise variables (23) are:

$\left\{\begin{array}{c}F_{2 k, \min }\left(z_{k}(t)\right)=\frac{z_{2 k}(t)-z_{2 k, \min }}{z_{2 k, \max }-z_{2 k, \min }} \\ F_{2 k, \max }\left(z_{k}(t)\right)=1-F_{2 k, \min }\left(z_{2 k}(t)\right)\end{array}\right.$    (24)

The global fuzzy model is inferred as follows:

$\dot{x_{2}}(t)=\sum_{i=1}^{8} \theta_{i}(z(t))\left(A_{2 i} x_{2}(t)+B_{2} u_{2}(t)+w_{2}(t)\right)$    (25)

where,

$\left\{\begin{aligned} \theta_{i}(z(t))=& \frac{\lambda_{i}(z(t))}{\sum_{i=1}^{8} \lambda_{i}(z(t))} ; \lambda_{i}(z(t))=\prod_{k=1}^{3} F_{2 i, k}\left(z_{2 k}(t)\right) \\ & \theta_{i}(z(t))>0 ; \sum_{i=1}^{8} \lambda_{i}(z(t))=1 \end{aligned}\right.$

$\theta_{i}(z(t))$ is the normalized weight of the i-rule, and $F_{2 i, k}\left(z_{2 k}(t)\right)$ is the grade of membership of $z_{2 k}(t)$ in $F_{2 i, k}$.

2.2.4 T-S fuzzy reference model for Induction Motor

Basing on the same previous principle, the T-S fuzzy reference model will generate the desired trajectory to be tracked by the state variables of the machine. A positive parameter K_i is inserted in the model, in order to improve the transient performance of the closed-loop system. The new control input vector is defined as follow [18]:

$\left\{\begin{array}{l}u_{s d c}^{\prime}=u_{s d c}+K_{i} \sigma L_{s}\left(i_{s d c}-i_{s d r}\right) \\ u_{s q c}^{\prime}=u_{s q c}+K_{i} \sigma L_{s}\left(i_{s q c}-i_{s q r}\right)\end{array}\right.$  (26)

$\left\{\begin{array}{c}u_{s d c}=\sigma L_{s}\left(\frac{d i_{s d c}}{d t}+\alpha i_{s d c}-w_{s c} i_{s q c}-\frac{K_{s}}{T_{r}} \psi_{d r c}\right) \\ u_{s q c}=\sigma L_{s}\left(\frac{d i_{s q c}}{d t}+\alpha i_{s q c}+w_{s c} i_{s d c}+K_{s} n_{p} w_{m c} \psi_{d r c}\right)\end{array}\right.$    (27)

The system (28) can be rewritten as:

$\left\{\begin{array}{c}u_{s d c}^{\prime}=\sigma L_{s}\left[\frac{d i_{s d c}}{d t}+\left(\alpha+K_{i}\right) i_{s d c}-w_{s c} i_{s q c}-\frac{K_{s}}{T_{r}} \psi_{d r c}-K_{i} \sigma L_{s} i_{s d r}\right] \\ u_{s q c}^{\prime}=\sigma L_{s}\left[\frac{d i_{s q c}}{d t}+\left(\alpha+K_{i}\right) i_{s q c}+w_{s c} i_{s d c}+K_{s} n_{p} w_{m c} \psi_{d r c} -K_{i} \sigma L_{s} i_{s q r}\right]\end{array}\right.$   (28)

Also,

$\left\{\begin{array}{l}u_{s d c}^{\prime}=u_{s d r}-K_{i} \sigma L_{s} i_{s d r} \\ u_{s q c}^{\prime}=u_{s q r}-K_{i} \sigma L_{s} i_{s q r}\end{array}\right.$   (29)

The state equation of the reference model of the IM is:

$\dot{x}_{2 r}(t)=A_{2 r} x_{2 r}(t)+r_{2}(t)$  (30)

where,

$\left(x_{2}(t)\right)=\left[\begin{array}{ccccc}-\xi_{1} & w_{s r} & \frac{K_{s}}{T_{r}} & K_{s} n_{p} w_{m r} & 0 \\ -w_{s r} & -\xi_{1} & -K_{s} n_{p} w_{m r} & \frac{K_{s}}{T_{r}} & 0 \\ \frac{M}{T_{r}} & 0 & -\frac{1}{T_{r}} & \frac{M}{T_{r} \psi_{r d c}} i_{s q r} & 0 \\ 0 & \frac{M}{T_{r}} & -\frac{M}{T_{r} \psi_{r d c}} i_{s q r} & -\frac{1}{T_{r}} & 0 \\ 0 & 0 & \frac{n_{p} M}{J_{r}} i_{s q r} & -\frac{n_{p} M}{J L_{r}} i_{s d r} & -\frac{f}{J}\end{array}\right]$

$\xi_{1}=\alpha+K_{i} ; w_{s r}=n_{p} w_{m r}+\frac{M}{T_{r} \psi_{r d c}} i_{s q r}$;

and $r_{2}(t)=\left[\begin{array}{cc}B_{2} & I\end{array}\right]\left[\begin{array}{c}u_{2 r} \\ w_{2}(t)\end{array}\right] ;$ where $u_{2 r}(t)=\left[\begin{array}{l}u_{s d r} \\ u_{s q r}\end{array}\right]$

The measurable premise variables are defined as:

$\left\{\begin{array}{l}z_{21 r}(t)=i_{s d r}(t) \\ z_{22 r}(t)=i_{s q r}(t) \\ z_{23 r}(t)=w_{m r}(t)\end{array}\right.$

The overall fuzzy reference model is given by:

$\dot{x}_{2 r}(t)=\sum_{j=1}^{8} \theta_{r j}\left(z_{2 r}(t)\right) A_{2 j r} x_{2 r}(t)+r_{2}(t)$    (31)

The H∞ performance related to the tracking error $e_{2 r}(t)=x_{2 r}(t)-x_{2}(t)$ is presented by the following inequality:

$\int_{0}^{t_{f}}\left[\left(x_{2 r}(t)-x_{2}(t)\right)^{T} Q_{2}\left(x_{2 r}(t)-x_{2}(t)\right)\right] d t \leq$

$\rho_{2}^{2} \int_{0}^{t_{f}}\left[\left(r_{2}(t)^{T} r_{2}(t)+w_{2}(t)^{T} w_{2}(t)\right] d t\right.$   (32)

r₂(t) is the reference input, w₂(t) is the external disturbance, Q₂ is a positive definite weighting matrix and ρ₂ is the attenuation level.

2.2.5 T-S fuzzy observer for Induction Motor

A Luenberger type multi-observer is constructed in this part to estimate the immeasurable state of the induction motor. Since the availability of all the variables is required, the T-S observer will to estimate the inaccessible rotor flux. As so, the T-S fuzzy observer is developed as so:

$\left\{\begin{array}{c}\dot{\hat{x}_{2}}(t)=\sum_{i=1}^{8} \theta_{i}(z(t))\left(A_{2 i} \hat{x}_{2}(t)+B_{2} u_{2}(t)\right. \\ \left.+L_{i}\left(y_{2}(t)-\hat{y}_{2}(t)\right)\right) \\ \hat{y}_{2}=C_{2} \hat{x}_{2}(t)\end{array}\right.$.    (33)

$\hat{x}_{2}(t)$ and $\hat{y}_{2}(t)$ designate respectively the estimated state and output vectors, while L_i are the observer gains.

We define the estimated error as $e_{2}(t)=x_{2}(t)-\hat{x}_{2}(t)$ for which the fuzzy model is given by:

$\dot{e}_{2}(t)=\sum_{i=1}^{8} \theta_{i}(z(t))\left[\left(A_{2 i}-L_{i} C_{2}\right) e_{2}(t)\right]+w_{2}(t)$   (34)

2.2.6 Tracking controller design

For the design of the observer-based controller, we adopted the same principle of the ordinary PDC controller but in addition, in the feedback control law, we take into account the reference state x_r(t) as following:

$u_{2}(t)=\sum_{i=1}^{8} \theta_{i}(z(t)) K_{2 i}\left(\hat{x}_{2}(t)-x_{2 r}(t)\right)$   (35)

where,  K_2i are the fuzzy control gain and $e_{2 r}(t)=\hat{x}_{2}(t)-x_{2 r}(t)$ is the tracking error for which the fuzzy model is as:

$\dot{e}_{2 r}(t)=\sum_{i=1}^{8} \sum_{j=1}^{8} \theta_{i}(z(t))\left(\theta_{j}\left(z_{2 r}(t)\right)\left[\left(A_{2 i}+\right.\right.\right.$

$\left.B_{2} K_{2 j}\right) e_{2 r}(t)-B_{2} K_{2 j} e_{2}(t)+\left(A_{2 i}-\right.$$\left.\left.A_{2 j r}\right) x_{2 r}(t)\right]+w_{2}(t)-r_{2}(t)$.    (36)

We obtain the following closed-loop system written in an augmented state-space form as:

$\dot{\bar{x}}_{2}(t)=\sum_{i=1}^{8} \sum_{j=1}^{8} \theta_{i}(z(t))\left(\theta_{j}\left(z_{2 r}(t)\right)\right.$ $\left.\times \overline{[G}_{i j} \bar{x}_{2}(t) \bar{E}_{2} \bar{w}_{2}(t)\right]$  (37)

$\bar{x}_{2}(t)=\left[\begin{array}{c}e_{2 r}(t) \\ e_{2}(t) \\ x_{2 r}(t)\end{array}\right] \quad \bar{w}_{2}(t)=\left[\begin{array}{c}w_{2}(t) \\ r_{2}(t)\end{array}\right] \quad \bar{E}_{2}=\left[\begin{array}{cc}I & -I \\ I & 0 \\ 0 & I\end{array}\right]$

$\bar{G}_{i j}=\left[\begin{array}{ccc}A_{2 i}+B_{2} K_{2 j} & -B_{2} K_{2 j} & A_{2 i}-A_{2 j r} \\ 0 & A_{2 i}-L_{i} C_{2} & 0 \\ 0 & 0 & A_{2 j r}\end{array}\right]$

Therefore, the H∞ tracking performance is reformulated as:

$\int_{0}^{t f} \overline{x_{2}}^{T}(t) \overline{Q_{2}} \overline{x_{2}}(t) d t \leq \rho_{2}^{2} \int_{0}^{t f} \overline{w_{2}}^{T}(t) \overline{w_{2}}(t) d t$    (38)

where, $\overline{Q_{2}}=\left[\begin{array}{lll}Q_{2} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]$

The next part is dedicated to study the stability of the closed loop system while maintaining a good trajectory tracking performance.

2.2.7 Reference speed

In order to search the optimum operating speed corresponding to the maximum power delivered by the PVG for various values of temperature and insolation, a reference speed bloc is added to compute instantly the reference speed $w_{m_{o p t}}$ that should be tracked and which assures the optimal operating of the IM, and thus a significant part of the available solar power is transferred to the unit motor-pump.

First, we start by defining the electrical power equation absorbed by the IM:

$P_{a b s}=\frac{3}{2} *\left(u_{s d} * i_{s d}+u_{s q} * i_{s q}\right)$    (39)

Then, and by replacing u_sd, u_sq,  u_sd, i_sd, and i_sq by their equations respectively (28) and (18), the DC electrical power will be expressed as a function of the motor speed in the following form as follow:

$P_{a b s}=C_{1}+C_{2} * w_{m o p t}^{2}+C_{3} * w_{m o p t}^{3}+C_{4} * w_{m o p t}^{4}$   (40)

$C_{1}=\sigma L_{S}\left(\frac{\psi_{r d}}{M}\right)^{2}\left(\alpha-K_{s} \frac{M}{T_{r}}\right)$

$C_{2}=\sigma L_{S}\left(\alpha K_{q}^{2} f^{2}+K_{s} K_{q} n_{p} f \psi_{r d}\right)$

$C_{3}=\sigma L_{S} A_{p}\left(2 \alpha K_{q}^{2} f+K_{s} K_{q} n_{p} \psi_{r d}\right)$

$C_{4}=\sigma L_{s} \alpha K_{q}^{2} A_{p}^{2} ; K_{q}=\frac{2 L_{r}}{3 n_{p} M \psi_{r d}}$

By assuming that the loses in DC converter is only in the inductance resistor R_L, and in the case of reaching the maximum power point conditions of the PVG, the optimal motor speed w_mopt can be obtained on-line by the Eq. (42) while adopting the following relation:

$P_{a b s}=P_{p v}-R_{L} * I_{p v}^{2}$    (41)

$w_{m_{o p t}}=\frac{P_{p v}-R_{L} * I_{p v}^{2}-C_{1}}{C_{2} * w_{m o p t}+C_{3} * w_{m o p t}^{2}+C_{4} * w_{m o p t}^{3}}$   (42)

2.3 Centrifugal pump

In this work, we used a centrifugal pump for which the characteristic H(Q) may be represented by:

$H=a Q^{2}+b Q w_{m}+c w_{m}^{2}$   (43)

H is the total head, Q designates the flow rate, w_m is the speed and a, b and c are three constants that depends on the pump’s dimensions. The above equation relates the main parameters of the centrifugal pump, the flow rate, the head and the speed. The pump performance is predicted by specifying a load curve [20]:

$H=H_{g}+\Delta H$   (44)

H_g is the static height (the distance between the free level of water and the highest point of the canalization), and ∆H is the pressure losses in the canalization equal to: 

$\Delta H=\frac{8\left(\frac{\partial l}{d}+\xi\right) Q^{2}}{\pi^{2} d^{4} g}$   (45)

where, ∂ is a resistance coefficient in the canalization, l is the length of the pipeline, d is the diameter of the pipeline, ξ is a local resistance coefficient and g is the gravity acceleration. Hence, the objective of this study is to determine a decentralised fuzzy controller with the H∞ tracking performance able to force the PV pumping system to operate very close to its maximum power trajectory for all climatic conditions’ variations. The main result is stated in the following two theorems.

Theorem 1: Considering the photovoltaic system (9). The closed-loop system (14) is asymptotically stable and the H∞ performance (15) with the attenuation level ρ is bounded, if there exist the following matrices $\mathrm{X}_{11}=\mathrm{P}_{11}^{-1}>0, \mathrm{P}_{12}>0, \mathrm{Y}_{1 \mathrm{i}}$  and a positive scalar ρ₁  solution for the following optimization problem:

$\min _{\left(P_{11}, P_{12}\right)}\left(\rho_{1}\right)$

Subject to:

$\left[\begin{array}{cccccccc}\psi_{11} & \left(A_{12}-A_{1 k r}\right) & E_{1} & -I & 0 & 0 & 0 & X_{11} \\ * & -2 \mu_{1} X_{11} & 0 & 0 & -\mu_{1} I & 0 & 0 & 0 \\ * & * & -2 \mu_{1} & 0 & 0 & -\mu_{1} I & 0 & 0 \\ * & * & * & -2 \mu_{1} & 0 & 0 & -\mu_{1} I & 0 \\ * & * & * & * & \psi_{22} & 0 & P_{12} & 0 \\ * & * & * & * & * & -\rho_{1}^{2} I & 0 & 0 \\ * & * & * & * & * & * & -\rho_{1}^{2} I & 0 \\ * & * & * & * & * & * & * & -Q_{1}^{-1}\end{array}\right]<0$    (46)

where,

$Y_{i}=K_{1 i} X_{11} ; \psi_{22}=A_{1 k r}^{T} P_{12}+P_{12} A_{1 k r}$

$\psi_{11}=A_{12} X_{11}+X_{11} A_{12}^{T}+B_{1 i} Y_{1 j}+Y_{1 j}^{T} B_{1 i}^{T}$

μ₁ is a scalar.

Theorem 2: For a positive scalar µ, the closed-loop system (37) is asymptotically stable and the H∞ performance (38) with the attenuation level ρ is bounded if there exist definite matrices $\mathrm{X}_{21}=\mathrm{P}_{21}^{-1}>0, \mathrm{P}_{22}>0, \mathrm{P}_{23}>0, \mathrm{Y}_{2 \mathrm{j}}, \mathrm{J}_{\mathrm{i}}$  and a positive scalar ρ₂  solution for the following optimisation problem:

$\min _{\left(P_{21}, P_{22}\right)}\left(\rho_{2}\right)$

Subject to:

$\left[\begin{array}{ccccccccc}\varphi_{11} & -B_{2} K_{2 j} & A_{2 i}-A_{2 j r} & I & -I & 0 & 0 & 0 & 0 & X_{21} \\ * & -2 \mu_{2} X_{21} & 0 & 0 & 0 & \mu_{2} I & 0 & 0 & 0 & 0 \\ * & * & -2 \mu_{2} I & 0 & 0 & 0 & \mu_{2} I & 0 & 0 & 0 \\ * & * & * & -2 \mu_{2} I & 0 & 0 & 0 & \mu_{2} I & 0 & 0 \\ * & * & * & * & -2 \mu_{2} I & 0 & 0 & 0 & \mu_{2} I & 0 \\ * & * & * & * & * & \varphi_{22} & 0 & P_{22} & 0 & 0 \\ * & * & * & * & * & * & \varphi_{33} & 0 & P_{23} & 0 \\ * & * & * & * & * & * & * & -\rho_{2}^{2} I & 0 & 0 \\ * & * & * & * & * & * & * & * & -\rho_{2}^{2} I & 0 \\ * & * & * & * & * & * & * & * & * & -Q_{2}^{-1}\end{array}\right]<0 $     (47)

where,

$Y_{2 j}=K_{2 j} X_{21}^{-1} ; L_{i}=P_{22}^{-1} J_{i}$

$\varphi_{11}=A_{2 i} X_{21}+X_{21} A_{2 i}^{T}+B_{2} Y_{2 j}+Y_{2 j}^{T} B_{2}^{T}$

$\varphi_{22}=A_{2 i}^{T} P_{22}+P_{22} A_{2 i}-J_{i} C_{2}-C_{2}^{T} J_{2 i}^{T}$

$\varphi_{33}=P_{23} A_{2 j r}+A_{2 j r}^{T} P_{23}$

and μ₂ is a scalar.

2.4 Global stability of PV pumping system

The stability of the global system is studied using the quadratic function of Lyapunov $V(\bar{x}(t))$ such that $\bar{P}_{i}$ is a positive definite symmetric matrix (i = 1,2) [19]:

$V(\bar{x}(t))=\bar{x}_{1}^{T}(t) \bar{P}_{1} \bar{x}_{1}(t)+\bar{x}_{2}^{T}(t) \bar{P}_{2} \bar{x}_{2}(t)$   (48)

The derivative of Lyapunov's function is written as:

$\dot{V}(\bar{x}(t))=\dot{\bar{x}_{1}^{T}}(t) \bar{P}_{1} \bar{x}_{1}(t)+\bar{x}_{1}^{T}(t) \bar{P}_{1} \dot {\bar{{x}_{1}}}(t)+\dot{\bar{x}_{2}^{T}}(t) \bar{P}_{2} \bar{x}_{2}(t)+\bar{x}_{2}^{T}(t) \bar{P}_{2} \dot{\bar{x}}_{2}(t)$  (49)

According to the H∞ tracking performances (15) and (38), and in order to realize the stability of the global pumping system, the following inequality must be verified:

$\dot{V}(\bar{x}(t))+\bar{x}_{1}^{T}(t) \bar{Q}_{1} \bar{x}_{1}(t)-\rho_{1}^{2} \bar{w}_{1}^{T}(t) \bar{w}_{1}(t)+\bar{x}_{2}^{T}(t) \bar{Q}_{2} \bar{x}_{2}(t)$

$-\rho_{2}^{2} \bar{w}_{2}^{T}(t) \bar{w}_{2}(t)<0$   (50)

Taking the derivative of V(x ̅(t))  with respect to t, the above inequality becomes:

$\sum_{i}^{4} \sum_{j}^{4} \sum_{k}^{2} h_{i j k}(z(t)) \sum_{i}^{8} \sum_{j}^{8} \mu_{i j}(z(t))\left[\begin{array}{l}\bar{x}_{1}(t) \\ \bar{w}_{1}(t) \\ \bar{x}_{2}(t) \\ \bar{w}_{2}(t)\end{array}\right]^{T} \times$

$\left[\begin{array}{cccc}{\bar{A}_{i j k}^{T} \bar{P}_{1}}+\bar{P}_{1} \bar{A}_{i j k}+\bar{Q}_{1} & \bar{P}_{1} \bar{E}_{1} & 0 & 0 \\ \bar{E}_{1}^{T} \bar{P}_{1} & -\rho_{1}^{2} I_{1} & 0 & 0 \\ 0 & 0 & \bar{G}_{i j}^{T} \bar{P}_{2}+\bar{P}_{2} \bar{G}_{i j}+\bar{Q}_{2} & \bar{P}_{2} \bar{E}_{2} \\ 0 & 0 & \bar{E}_{2}^{T} \bar{P}_{2} & -\rho_{2}^{2} I_{2}\end{array}\right]\times\left[\begin{array}{l}\bar{x}_{1}(t) \\ \bar{w}_{1}(t) \\ \bar{x}_{2}(t) \\ \bar{w}_{2}(t)\end{array}\right]<0$  (51)

We note that $\begin{array}{lll} \bar{P}_{1}=\operatorname{diag}\left[P_{11} P_{12}\right] & \text { and } & \bar{P}_{2}=\end{array}\operatorname{diag}\left[P_{21} P_{22} P_{23}\right] .$ Which allows to extract the following two inequalities:

$\left[\begin{array}{cc}\bar{A}_{i j k}^{T} \bar{P}_{1}+\bar{P}_{1} \bar{A}_{i j k}+\bar{Q}_{1} & \bar{P}_{1} \bar{E}_{1} \\ \bar{E}_{1}^{T} \bar{P}_{1} & -\rho_{1}^{2} I_{1}\end{array}\right]<0$   (52)

$\left[\begin{array}{cc}\bar{G}_{i j}^{T} \bar{P}_{2}+\bar{P}_{2} \bar{G}_{i j}+\bar{Q}_{2} & \bar{P}_{2} \bar{E}_{2} \\ \bar{E}_{2}^{T} \bar{P}_{2} & -\rho_{2}^{2} I_{2}\end{array}\right]<0$   (53)

By replacing each augmented matrix with its expression, the above inequalities are bilinear matrix inequalities. So, to resolve this problem, it is necessary to pre-and post-multiply the $\mathrm{BMIs}$ by $Z_{i}=\operatorname{diag}\left[P_{i 1}^{-1} W_{i}\right](\mathrm{i}=1,2),$ where $W_{1}=$$\operatorname{diag}\left[P_{11}^{-1} I I\right]$ and $W_{2}=\operatorname{diag}\left[P_{21}^{-1} I I I\right] ;$ Which leads to obtain an inequality in the form:

$\left[\begin{array}{cc}P_{i 1}^{-1} \Pi_{i 11} P_{i 1}^{-1} & P_{i 1}^{-1} \Pi_{i 12} W_{i} \\ * & W_{i} \Pi_{i 22} W_{i}\end{array}\right]<0$  (54)

The verification of this condition will signify that $W_{i} \Pi_{i 22} W_{i}<0$ so the above inequality is equivalent to [7]:

$\left(W_{i}+\mu_{i} \Pi_{i 22}^{-1}\right)^{T} \Pi_{i 22}\left(W_{i}+\mu_{i} \Pi_{i 22}^{-1}\right)<0 \Leftrightarrow$

$W_{i} \Pi_{i 22} W_{i}<-2 \mu_{i} W_{i}-\mu_{i}^{2} \Pi_{i 22}^{-1}$   (55)

where, μ_i is a scalar.

By substituting (55) to (54) and applying the Schur complement, we get:

$\left[\begin{array}{ccc}P_{i 1}^{-1} \Pi_{i 11} P_{i 1}^{-1} & P_{1}^{-1} \Pi_{i 12} W_{i} & 0 \\ * & -2 \mu_{i} W_{i} & -\mu_{i} I_{i} \\ * & * & \Pi_{i 22}\end{array}\right]<0$   (56)

By applying the Schur complement one second time for both inequalities (i=1, 2), we obtain two standard LMIs mentioned in the above two theorems which the resolution of each one gives the control gains for each subsystem.