Intelligent Control of Wind Energy Conversion Systems

1. Introduction

Wind turbines form complex nonlinear mechanical systems exposed to uncontrolled wind profiles. This makes turbine controller design a challenging task (Athanasius & Zhu, 2009). As such, control of wind energy conversion systems (WECS) is difficult due to the lack of systematic methods to identify requisite robust and sufficiently stable conditions, to guarantee performance. The problem becomes more complex when plant parameters become uncertain. Fuzzy control is one of the techniques which deal with this class of systems. The stability of fuzzy systems formed by a fuzzy plant model and a fuzzy controller has recently been investigated. Various stability conditions have been obtained through the employment of Lyapunov stability theory (Schegner & La Seta, 2004; Tripathy, 1997), fuzzy gain- scheduling controllers (Billy, 2011a, 2011b; Iescher et al., 2005), switching controllers (Lescher et al., 2006) and by other methods (Chen & Hu, 2003; Kamal et al., 2008; Muljadi & Edward, 2002). Nonlinear controllers (Boukhezzar & Siguerdidjane, 2009; Chedid et al., 2000; Hee-Sang et al., 2008) have also been proposed for the control of WECS represented by fuzzy models.

In addition to stability, robustness is also an important requirement to be considered in the study of uncertain nonlinear WECS control systems. Robustness in fuzzy-model-based control has been extensively studied, such as stability robustness versus modelling errors and other various control techniques for Takagi–Sugeno (TS) fuzzy models (Kamal et al., 2010; Uhlen et al., 1994).

In order to overcome nonlinearity and uncertainties, various schemes have been developed in the past two decades (Battista & Mantz, 2004; Boukhezzar & Siguerdidjane, 2010; Prats et al., 2000; Sloth et al., 2009). (Battista & Mantz, 2004) addressing problems of output power regulation in fixed-pitch variable-speed wind energy conversion systems with parameter uncertainties. The design of LMI-based robust controllers to control variable-speed, variable-pitch wind turbines, while taking into account parametric uncertainties in the aerodynamic model has been presented (Sloth et al., 2009). (Boukhezzar & Siguerdidjane, 2010) comparing several linear and nonlinear control strategies, with the aim of improving wind energy conversion systems. (Prats et al., 2000) have also investigated fuzzy logic controls to reduce uncertainties faced by classical control methods.

Furthermore, although the problem of control in the maximization of power generation in variable-speed wind energy conversion systems (VS-WECS) has been greatly studied, such control still remains an active research area (Abo-Khalil & Dong-Choon, 2008; Aggarwal et al., 2010; Barakati et al., 2009; Camblong et al., 2006; Datta & Ranganathan, 2003; Galdi et al., 2009; Hussien et al., 2009; Iyasere et al., 2008; Koutroulis & Kalaitzakis, 2006; Mohamed et al., 2001; Prats et al., 2002; Whei-Min. & Chih-Ming, 2010). (Abo-Khalil & Dong-Choon, 2008; Aggarwal et al., 2010; Camblong et al., 2006; Datta & Ranganathan, 2003; Whei-Min. & Chih-Ming, 2010) maximum power point tracking (MPPT) algorithms for wind turbine systems have been presented (Galdi et al., 2009) as well as design methodology for TS fuzzy models. This design methodology is based on fuzzy clustering methods for partitioning the input-output space, combined with genetic algorithms (GA), and recursive least-squares (LS) optimization methods for model parameter adaptation. A maximum power tracking algorithm for wind turbine systems, including a matrix converter (MC) has been presented (Barakati et al., 2009). A wind-generator (WG) maximum-power-point tracking (MPPT) system has also been presented (Koutroulis & Kalaitzakis, 2006), consisting of a high efficiency buck-type dc/dc converter and a microcontroller-based control unit running the MPPT function. An advanced maximum power-tracking controller of WECS (Mohamed et al., 2001), achieved though the implementation of fuzzy logic control techniques, also appears promising. The input to the controller consists in the difference between the maximum output power from the WES and the output power from the asynchronous link and, the derivative of this difference. The output of the controller is thus the firing angle of the line-commutated inverter, which transfers the maximum tracked power to the utility grid. Fuzzy controllers also permit the increase of captured wind energy under low and high wind speeds (Prats et al., 2002; Hussien et al., 2009). The fuzzy controller is employed to regulate, indirectly, the power flow in the grid connected WECS by regulating the DC current flows in the interconnected DC link. Sufficiently stable conditions are expressed in terms of Linear Matrix inequalities (LMI). (Iyasere et al., 2008) to maximize the energy captured by the wind turbine under low to medium wind speeds by tracking the desired pitch angle and rotor speed, when the wind turbine system nonlinearities structurally uncertain.

Concerning other studies, due to the strong requirements of the Wind Energy Field, fault tolerant control of variable speed wind turbine systems has received significant attention in recent years (Bennouna et al., 2009; Gaillard et al., 2007; Odgaard et al., 2009; Ribrant, 2006; Wang et al., 2010; Wei et al., 2010). To maintain the function of closed-loop control during faults and system changes, it is necessary to generate information about changes in a supervision scheme. Therefore, the objective of Fault Tolerant Control (FTC) is to maintain current performances close to desirable performances and preserve stability conditions in the presence of component and/or instrument faults. FTC systems must have the ability to adjust off-nominal behaviour, which might occur during sensor, actuator, or other component faults. A residual based scheme has been presented (Wei et al., 2010) to detect and accommodate faults in wind turbines. An observer based scheme (Odgaard et al., 2009) has been proposed to detect and isolate sensor faults in wind turbine drive trains. A study of fault tolerant power converter topology (Gaillard et al., 2007) and fault identification and compensation for a WECS with doubly fed induction generator (DFIG), has also been done. In addition, a survey on failures of wind turbine systems in Sweden, Finland and Germany (Ribrant, 2006), has been carried out, where the data are from real maintenance records over the last two decades. Robust fault tolerant controllers based on the two-frequency loop have also been designed (Wang et al., 2010). The low-frequency-loop adopts a PI steady-state optimization control strategy, and the high-frequency-loop adopts a robust fault tolerant control approach, thus ensuring the actuator part of the system during failure in normal operation. Fault signature analysis to detect errors in the DFIG of a wind turbine has again been presented (Bennouna et al., 2009).

It is well known that observer based design is a very important problem in control systems. Since in many practical nonlinear control systems, state variables are often unavailable, output feedback or observer-based control is necessary and these aspects have received much interest. (Khedher et al.,2009, 2010; Odgaard et al., 2009; Tong & Han-Hiong, 2002; Tong et al., 2009 ; Wang et al., 2008; Yong-Qi, 2009; Zhang et al., 2009) fuzzy observer designs for TS fuzzy control systems have been studied, and prove that a state feedback controller and observer always result in a stabilizing output feedback controller, provided that the stabilizing property of the control and asymptotic convergence of the observer are guaranteed through the Lyapunov method. However, in the above output feedback fuzzy controllers, the parametric uncertainties for TS fuzzy control systems have not been considered. As such robustness of the closed-loop system may not be guaranteed.

In this chapter, a Robust Fuzzy Fault Tolerant control (RFFTC) algorithm is proposed for hybrid wind-diesel storage systems (HWDSS) with time-varying parameter uncertainties, sensor faults and state variable unavailability, and measurements based on the Takagi-Sugeno (TS) fuzzy model. Sufficient conditions are derived for robust stabilization in the sense of Lyapunov asymptotic stability and are formulated in the form of Linear Matrix Inequalities (LMIs). The proposed algorithm combines the advantages of:

• The capability of dealing with non-linear systems with parametric uncertainties and sensor faults;

• The powerful Linear Matrix Inequalities (LMIs) approach to obtain fuzzy fault tolerant controller gains and observer gains;

• The maximization of the power coefficient for variable pitch variable-speed wind energy conversion systems;

• In addition, reduction of voltage ripple and stabilization of the system over a wide range of sensor faults and parameter uncertainties is achieved.

Also in this chapter, a Fuzzy Proportional Integral Observer (FPIO) design is proposed to achieve fault estimation in TS fuzzy models with sensor faults and parameter uncertainties. Furthermore, based on the information of online fault estimation, an observer-based robust fuzzy fault tolerant controller is designed to compensate for the effects of faults and parameter uncertainties, by stabilizing the closed-loop system. Based on the aforementioned studies, the contributions of this chapter are manifold:

• A new algorithm for the estimation of time-varying process faults and parameter uncertainties in a class of WECS;

• And a composite fault tolerant controller to compensate for the effects of the faults, by stabilizing the closed-loop system in the presence of bounded time-varying sensor faults and parameter uncertainties.

This chapter is organized as follows. In section 2, the dynamic modelling of WECS and system descriptions is introduced. Section 3 describes the fuzzy plant model, the fuzzy observer and the reference model. In section 4, robust fuzzy fault tolerant algorithms are proposed, to close the feedback loop and the stability and robustness conditions for WECS are derived and formulated into nonlinear matrix inequality (general case) and linear matrix inequality (special case) problems. Section 5 presents the TS Fuzzy Description and Control structure for HWDSS. Section 6 summarizes the procedures for finding the robust fuzzy fault tolerant controller and fuzzy observer. In section 7 simulation results illustrate the effectiveness of the proposed control methods for wind systems. In section 8, a conclusion is drawn.

2. WECS model and systems descriptions

2.1. The wind turbine characteristics

Variable Speed wind turbine has three main regions of operation as shown in Fig.1. (Galdi et al., 2009). The use of modern control strategies are not usually critical in region I, where the monitoring of the wind speed is performed to determine whether it lies within the specifications for turbine operation and if so, the routines necessary to start up the turbine are performed. Region II is the operational mode in which the goal is to capture as much power as possible from the wind. Region III is called rated wind speed. The control objectives on the full load area are based on the idea that the control system has to maintain the output power value to the nominal value of the generator. The torque at the turbine shaft neglecting losses in the drive train is given by (Iyasere et al., 2008):

$$T_G=0.5\pi C_t(\lambda ,\beta ){{\displaystyle R}}^3\rho {\nu }^2$$ E1

where T _G is the turbine mechanical torque, Where ρ is the air density (kg /m³), R is the turbine radius (m),ν is the wind velocity (m/s), and C _t (λ,β) is the turbine torque coefficient. The power extracted from the wind can be expressed as (Galdi et al., 2009) :

$$P_a={\omega }_t{T}_G=0.5C_p(\lambda ,\beta )\rho \pi {{\displaystyle R}}^2{{\displaystyle \nu }}_{}^3$$ E2

where C _p (λ,β) is the rotor power coefficient defined by the following relation,

$$C_p(\lambda ,\beta )=\lambda C_t(\lambda ,\beta )$$ E3

$${{\displaystyle C}}_p(\lambda ,\beta )=(0.44-0.0167\beta )\sin [\pi (\lambda -3)/15-0.3\beta ]-0.00184(\lambda -3)\beta$$ E4

β is the pitch angle of rotor blades (rad) (β is constant for fixed pitch wind turbines), λ is the tip speed ratio (TSR) and is given by:

$$\lambda =R{\omega }_t/\nu$$ E5

where $${\omega }_t$$ is the rotor speed (rad/sec). It is seen that if the rotor speed is kept constant, then any change in the wind speed will change the tip-speed ratio, leading to the change of power coefficient C _p as well as the generated power out of the wind turbine. If, however, the rotor speed is adjusted according to the wind speed variation, then the tip-speed ratio can be maintained at an optimal point, which could yield maximum power output from the system. Referring to (3) optimal TSR λ _opt can be obtained as follow:

$${{\displaystyle \lambda }}_{opt}=(\frac{15-0.3\beta }{\pi }){{\displaystyle \cos }}^{-1}[\frac{0.00184\beta (15-0.3\beta )}{\pi (0.44-0.167\beta )}]+3$$ E6

From (5) it is clear that λ _opt depends on β. The relationship of C _p versus λ, for different values of the pitch angle β, are shown in Fig 2. The maximum value of C _p (C _p(max) = 0.48) is achieved for β = 0^o and for λ= 8. This particular value of λ is defined as the optimal value of TSR (λ _opt ).Thus the maximum power captured from the wind is given by:

$$P_{a(\max )}=0.5C_{p(\max )}({{\displaystyle \lambda }}_{opt},\beta )\rho \pi {{\displaystyle R}}^2{{\displaystyle \nu }}^3$$ E7

Normally, a variable speed wind turbine follows the C _p(max) to capture the maximum power up to the rated speed by varying the rotor speed at ω _opt to keep the TSR at λ _opt .

Figure 1.

Power-wind speed characteristics

2.2. WECS system description

A wind-battery hybrid system consists of a wind turbine coupled with a synchronous generator (SG), ), a diesel-induction generator (IG) and a battery connected with a three-phase thyristor-bridge controlled current source converter. In the given system, the wind turbine drives the synchronous generator that operates in parallel with the storage battery system. When the wind-generator alone provides sufficient power for the load, the diesel engine is disconnected from the induction generator. The Power Electronic Interface (PEI) connecting the load to the main bus is used to fit the frequency of the power supplying the load as well as the voltage. Fig. 3 shows the overall structure of wind-battery system: E _fd is the excitation field voltage, f is the frequency, V _b is the bus voltage, C _a is the capacitor bank, V _c is the AC side voltage of the converter, and I _ref is the direct-current set point of the converter.

Figure 2.

Power coefficient C _p versus TSR λ

The dynamics of the system can be characterized by the following equations (Kamal et al., 2010):

$$\dot{x}=Ax(t)+Bu(t),y=Cx(t)$$ E8

where

$$x\text{(t) }={\text{[V}{}_{\text{b}}{\omega }_{\text{s}}\text{]}}^{\text{T}},u\text{(t) }={\text{[E}{}_{\text{fd}}{I}_{\text{ref}}\text{]}}^{\text{T}}$$ E9

$$A=\left[\begin{array}{l}\text{1}1\\ \text{0}1\end{array}\right]\left[\begin{array}{l}\frac{{{\displaystyle L}}_f}{{{\tau }^{\prime }}_{}{{\displaystyle \omega }}_s{{\displaystyle L}}_{md}}\frac{{{\displaystyle L}}_f}{{{\tau }^{\prime }}_{}{{\displaystyle {{\displaystyle \omega }}_{s}L}}_{md}}({{\displaystyle L}}_d{{\displaystyle i}}_{sd-\frac{{{\displaystyle {{\displaystyle r}}_{a}i}}_{sq}}{{{\displaystyle \omega }}_s}})\\ \frac{{{\displaystyle {{\displaystyle P}}_{ind}-P}}_{load}}{{{\displaystyle {{\displaystyle {{\displaystyle J}}_s\omega }}_{s}V}}_b}-\frac{{{\displaystyle D}}_s}{{{\displaystyle J}}_s}\end{array}\right]B=\left[\begin{array}{l}\text{1}-\frac{{{\displaystyle V}}_c}{{{\displaystyle {{\displaystyle \omega }}_{s}J}}_s}\\ \text{0}-\frac{{{\displaystyle V}}_c}{{{\displaystyle {{\displaystyle \omega }}_{s}J}}_s}\end{array}\right]\text{ ,C}=\left[\begin{array}{l}\text{1}\text{0}\\ \text{0}\text{1}\end{array}\right]$$ E10

where ω _s is the bus frequency (or angular speed of SG) J _s , D _s are the inertia and frictional damping of SG, i _sd , i _sq are the direct and quadrature current component of SG, L _d , L _f are the stator d-axis and rotor inductance of SG, L _md is the d-axis field mutual inductance, τ`is the transient open circuit time constant, r _a is the rotor resistance of SG, P _ind is the power of the induction generator, P _load is the power of the load. Equation (7) indicates that the matrices A and B are not fixed, but change as functions of state variables, thus making the model nonlinear. Also, this model is only used as a tool for controller design purposes.

Figure 3.

Structural diagram of hybrid wind-diesel storage system

The used system parameters are shown in Table 1 (Chedid et al., 2000; Kamal et al., 2010).

Rated power	1 [MW]
Blade radius	37.38 [m]
Air density	0.55 [kg /m^3]
Rated wind speed	12.35 [m/s]
Blade pitch angle	0^o
Rated line ac voltage	230 [V]
AC rated current	138 [A]
DC rated current	239 [A]
Rated Load power	40 [kW]
The inertia of SG	1.11 [kg m^2]
Rated power of IG	55 [kW]
The inertia of the IG	1.40 [kg m^2]
Torsional damping	0.557 [Nm/ rad]
Rotor resistance of SG	0.96 [ Ω]
Stator d-axis inductance of SG	2.03 [mH]
Rotor inductance of SG	2.07 [mH]
d-axis field mutual inductance	1.704 [mH]
The transient open circuit time constant	2.16 [ms]

Table 1.

System parameters

3. Reference model, TS fuzzy plant model and fuzzy proportional-integral observer

3.1. Reference model

A reference model is a stable linear system without faults given by (Khedher et al., 2009, 2010),

$$\dot{\overline{x}}(t)={\text{A}}_{\text{r}}\text{ <mover accent="true"><mi>x</mi> <mo>¯</mo> </mover>(t)}+{\text{B}}_{\text{r}}\text{r(t),}\overline{y}(t)=C_r\overline{x}(t)$$ E11

where $$\text{<mover accent="true"><mi>x</mi> <mo>¯</mo> </mover>}(t)\in {\kappa }^{\text{nx1}}$$ is the state vector of reference model, $$\text{r}(t)\in {\kappa }^{\text{nx1}}$$ is the bounded reference input, $$A_r\in {\kappa }^{\text{nxn}}$$ is the constant stable system matrix, $$B_r\in {\kappa }^{\text{nxn}}$$ is the constant input matrix, $$C_r\in {\kappa }^{\text{gxn}}$$ is the constant output matrix. $$\text{<mover accent="true"><mi>y</mi> <mo>¯</mo> </mover>}(t)\in {\kappa }^{\text{gx1}}$$ is the reference output.

3.2. TS fuzzy plant model with parameter uncertainties and sensor faults

The continuous fuzzy dynamic model, proposed by TS, is described by fuzzy IF-THEN rules, which represent local linear input-output relations of nonlinear systems. Consider an uncertain nonlinear system that can be described by the following TS fuzzy model with parametric uncertainties and sensor faults (Khedher et al., 2009, 2010; Tong & Han-Hiong, 2002). The i-th rule of this fuzzy model is given by:

Plant Rule i: q ₁ (t) is N ⁱ ₁ AND … AND q _ψ (t) is N ⁱ _ψ

Then

$$\begin{array}{l}\dot{x}(t)=({\text{A}}_{\text{i}}+\Delta {\text{A}}_{\text{i}}\text{) x(t)}+{\text{B}}_{\text{i}}\text{u(t),}\\ y(t)=C_{i}x(t)+{\text{E}}_{\text{i}}\text{f(t)}\end{array}$$ E12

where N ⁱ _Ω is a fuzzy set of rule i, Ω = 1,2,…,ψ, i=1,2,…,p, $$\text{x}(t)\in {\kappa }^{\text{nx1}}$$ is the state vector, $$\text{u}(t)\in {\kappa }^{\text{nx1}}$$ is the input vector, $$\text{y}(t)\in {\kappa }^{\text{gx1}}$$ is the output vector, $$\text{f}(t)\in {\kappa }^{\text{mx1}}$$ represents the fault which is assumed to be bounded, $$A_i\in {\kappa }^{\text{nxn}}$$ and $$B_i\in {\kappa }^{\text{nxn}}$$ , $$C_i\in {\kappa }^{\text{gxn}}$$ , $${\text{E}}_{\text{i}}$$ are system matrix, input matrix, output matrix and fault matrix, respectively, which are assumed to be known, $$\Delta A_i\in {\kappa }^{\text{nxn}}$$ is the parameter uncertainties of $$A_i$$ within known. It is supposed that the matrix E _i is of full column rank, i.e. rank(E _i )= r. It is assumed that the derivative of f(t) with respect to time is norm bounded, i.e. $$\Vert \text{<mover accent="true"> <mi>f</mi> <mo>˙</mo> </mover>}(t)\Vert \le f_1$$ and $$0\le f_1\infty$$ (Zhang et al., 2009), p is the number of IF-THEN rules, and q ₁ (t),…, q _ψ (t) are the premise variables assumed measurable variables and do not depend on the sensor faults. The defuzzified output of (9) subject to sensor faults and parameter uncertainty is represented as follows (Khedher et al., 2009, 2010; Tong & Han-Hiong, 2002):

$$\begin{array}{l}\dot{x}(t)={\displaystyle \sum _{\text{i}=\text{1}}^{\text{p}}{{\displaystyle \mu }}_i\text{(q(t))[}({\text{A}}_{\text{i}}+\Delta {\text{A}}_{\text{i}}\text{)x(t)}+B_i\text{U(t)] }}\text{ ,}\\ y(t)={\displaystyle \sum _{\text{i}=\text{1}}^{\text{p}}{{\displaystyle \mu }}_i\text{(q(t))[}{C}_{\text{i}}\text{x(t)}+{\text{E}}_{\text{i}}\text{f(t)] }}\end{array}$$ E13

where

$$\begin{array}{l}{h}_i(q(t))=\underset{\alpha =1}{\overset{\psi }{{\displaystyle \Pi }}}{N}_{\alpha }^i(q(t)),\\ {\mu }_{\text{i}}\text{(q(t))}=h_i(q(t))\text{ /}{\displaystyle \sum _{i=1}^p{h}_i(q(t))}\end{array}$$ E14

Some basic properties of

$$\begin{array}{l}{h}_i(q(t))\text{ are,}{h}_i(q(t))\ge \text{0}\\ {\displaystyle \sum _{\text{i}=\text{1}}^{\text{p}}{h}_i(q(t))\ge \text{0 }}0\le {\mu }_i\text{(q(t))}\le 1,\\ {\displaystyle \sum _{\text{i}=\text{1}}^{\text{p}}{\mu }_{\text{i}}\text{(q(t))}=\text{1 }}\forall \text{i}=\text{1,2,}\mathrm{....}\text{,p }\end{array}$$ E15

3.3. TS Fuzzy Proportional Integral Observer (FPIO)

Definition 1: If the pairs (A _i ,C _i ), i=1, 2,…,p, are observable, the fuzzy system (10) is called locally observable (Xiao-Jun et al., 1998).

For the fuzzy observer design, it is assumed that the fuzzy system (10) is locally observable. First, the local state observers are designed as follows, based on the triplets (A _i ,B _i ,,C _i ). In order to detect and estimate faults, the following fault estimation observer is constructed (Khedher et al., 2009, 2010; Tong & Han-Hiong, 2002).

Observer Rule i: q ₁ (t) is N ⁱ ₁ AND … AND q _ψ (t) is N ⁱ _ψ

Then

$$\begin{array}{l}\dot{\widehat{x}}(t)=A_i\text{<mover accent="true"> <mi>x</mi> <mo>^</mo> </mover>(t)}+B_i\text{u(t)}+{\text{K}}_{\text{i}}\text{(y(t)-<mover accent="true"> <mi>y</mi> <mo>^</mo> </mover>(t)),}\\ \text{<mover accent="true"> <mover accent="true"> <mi>f</mi> <mo>^</mo> </mover> <mo>˙</mo> </mover>(t)}={\text{L}}_{\text{i}}\text{(y-<mover accent="true"> <mi>f</mi> <mo>^</mo> </mover>)}={\text{L}}_{\text{i}}\text{<mover accent="true"> <mi>f</mi> <mo>˜</mo> </mover>}\\ \widehat{y}(t)=C_i\widehat{x}(t)+{\text{E}}_{\text{i}}\text{<mover accent="true"> <mi>f</mi> <mo>^</mo> </mover>(t),}\\ i=1,2,\dots ,p\end{array}$$ E16

where K _i is the proportional observer gain for the i-th observer rule and L _i are their integral gains to be determined, $$y(t)$$ and $$\widehat{y}(t)$$ are the final output of the fuzzy system and the fuzzy observer respectively. The defuzzified output of (12) subject to sensor faults is represented as follows:

$$\begin{array}{l}\dot{\widehat{x}}(t)={\displaystyle \sum _{\text{i}=\text{1}}^{\text{p}}{{\displaystyle \mu }}_i{\text{(q(t))[A}}_{\text{i}}\text{ <mover accent="true"> <mi>x</mi> <mo>^</mo> </mover>(t)}+B_i\text{U(t)}+{\text{K}}_{\text{i}}\text{(y(t)-<mover accent="true"> <mi>y</mi> <mo>^</mo> </mover>(t))] }}\\ \text{ <mover accent="true"> <mover accent="true"> <mi>f</mi> <mo>^</mo> </mover> <mo>˙</mo> </mover>(t)}={\displaystyle \sum _{i=1}^p{\mu }_i}{\text{L}}_{\text{i}}\text{(y-<mover accent="true"> <mi>y</mi> <mo>^</mo> </mover>)}={\displaystyle \sum _{i=1}^p{\mu }_i}{\text{L}}_{\text{i}}\text{<mover accent="true"> <mi>y</mi> <mo>˜</mo> </mover>}\\ \widehat{y}(t)={\displaystyle \sum _{\text{i}=\text{1}}^{\text{p}}{{\displaystyle \mu }}_i\text{(q(t))[}{C}_{\text{i}}\text{<mover accent="true"> <mi>x</mi> <mo>^</mo> </mover>(t) }+{\text{E}}_{\text{i}}\text{<mover accent="true"> <mi>f</mi> <mo>^</mo> </mover>(t)]}}\end{array}$$ E17

The goal is to design the control law u(t) such that the system state x(t) will follow those of a stable reference model of (8) in the presence of parametric uncertainties and sensor faults.

4.1. PDC technique

The concept of PDC in (Wang et al., 1996) is utilized to design fuzzy controllers to stabilize fuzzy system (10). For each rule, we can use linear control design techniques. The resulting overall fuzzy controller, which is nonlinear in general, is a fuzzy blending of each individual linear controller. The fuzzy controller shares the same fuzzy sets with the fuzzy system (10).

4.2. Proposed RFFTC controller

Definition 2: If the pairs (A _i , B _i ), i= 1, 2,…, p, are controllable, the fuzzy system (10) is called locally controllable (Xiao-Jun et al., 1998).

For the fuzzy controller design, it is assumed that the fuzzy system (10) is locally controllable. First, the local state feedback controllers are designed as follows, based on the pairs (A _i ,B _i ). Using PDC the i-th rule of the fuzzy controller is of the following format:

Controller Rule i: q ₁ (t) is N ⁱ ₁ AND … AND q _ψ (t) is N ⁱ _ψ

Then

$$u\left(t\right)=u_i\left(t\right)$$ E18

where $${\text{u}}_{\text{i}}(t)\in {\kappa }^{\text{nx1}}$$ is the output of the i-th rule controller that will be defined in the next sub-section. The global output of the fuzzy controller is given by

$$U(t)={\displaystyle \sum _{\text{i}=\text{1}}^{\text{p}}{{\displaystyle \mu }}_i{\text{(q(t))u}}_{\text{i}}\text{(t) }}$$ E19

From (10), (11) and (15), writing $${{\displaystyle \mu }}_i\text{(q(t))}$$ as $${{\displaystyle \mu }}_i$$ , we obtain

$$\begin{array}{l}\dot{x}(t)={\displaystyle \sum _{\text{i}=\text{1}}^{\text{p}}{{\displaystyle \mu }}_i({\text{A}}_{\text{i}}+\Delta {\text{A}}_{\text{i}}\text{) x(t)}+B{u}_i\text{ ] }}\\ y(t)={\displaystyle \sum _{\text{i}=\text{1}}^{\text{p}}{{\displaystyle \mu }}_i{{\displaystyle \text{[}C}}_i\text{ x(t)}+\text{Ef(t) ] }}\end{array}$$ E20

where

$$\begin{array}{l}B={\displaystyle \sum _{\text{i}=\text{1}}^{\text{p}}{{\displaystyle \mu }}_i}{B}_i\\ and\\ E={\displaystyle \sum _{\text{i}=\text{1}}^{\text{p}}{{\displaystyle \mu }}_i}{E}_i\end{array}$$ E21

Note that B and E are known. Also from (11), (12) and (15), we have

$$\begin{array}{l}\dot{\widehat{x}}(t)={\displaystyle \sum _{\text{i}=\text{1}}^{\text{p}}{{\displaystyle \mu }}_i\text{[}{A}_{\text{i}}\text{<mover accent="true"> <mi>x</mi> <mo>^</mo> </mover>(t)}+B{u}_i(t)+{\text{K}}_{\text{i}}\text{(y(t)-<mover accent="true"> <mi>y</mi> <mo>^</mo> </mover>(t))] }}\\ \text{ <mover accent="true"> <mover accent="true"> <mi>f</mi> <mo>^</mo> </mover> <mo>˙</mo> </mover>(t)}={\displaystyle \sum _{i=1}^p{\mu }_i}{\text{L}}_{\text{i}}\text{(y-<mover accent="true"> <mi>y</mi> <mo>^</mo> </mover>)}={\displaystyle \sum _{i=1}^p{\mu }_i}{\text{L}}_{\text{i}}\text{<mover accent="true"> <mi>y</mi> <mo>˜</mo> </mover>}\\ \widehat{y}(t)={\displaystyle \sum _{\text{i}=\text{1}}^{\text{p}}{{\displaystyle \mu }}_i\text{[}{C}_{\text{i}}\text{<mover accent="true"> <mi>x</mi> <mo>^</mo> </mover>(t) }+\text{E<mover accent="true"> <mi>f</mi> <mo>^</mo> </mover>(t)]}}\end{array}$$ E22

We derived the stability and robustness conditions for a generalized class of WECS with sensor faults and parameter uncertainties described by (16).The main result is summarized in the following lemma 1 and theorem.

Lemma 1: The fuzzy control system of (16) subject to plant sensors faults and parameter uncertainties is guaranteed to be asymptotically stable, and its states will follow those of a stable reference model of (8) in the presence of bounded sensor faults and parameter uncertainties, if the following two conditions satisfy;

• Z is nonsingular. One sufficient condition to guarantee the nonsingularity of Z is that there exists P such that,

• The control laws of fuzzy controller of (15) are designed as, If e ₁ (t)≠0; When B is an invertible square matrix, the control law is given by

$$u_i(t)=Z^{-1}{Z}_{ui}$$ E24

When B is not a square matrix, the control law is given by

$$u_i(t)=B^T{Z}^{-1}{Z}_{ui}$$ E25

Where Z _ui is given by

$$\begin{array}{l}{Z}_{ui}=\{{\text{[He}}_{\text{1}}\text{(t)}+{\text{A}}_{\text{r}}\text{ <mover accent="true"> <mi>x</mi> <mo>¯</mo> </mover>(t)}+{\text{B}}_{\text{r}}\text{r(t)}-\frac{{\text{e}}_{\text{1}}(t)\Vert {\text{e}}_{\text{1}}(t)\Vert \Vert P_1\Vert {\Vert \Delta {\text{A}}_{\text{i}}\Vert }_{\max }\Vert \text{x(t)}\Vert }{{\text{e}}_{\text{1}}{(t)}^T{P}_1{\text{e}}_{\text{1}}(t)}\\ -\frac{{\text{e}}_{\text{1}}(t)\Vert \text{x(t)}\Vert {\Vert D\Vert }_{\max }\Vert \text{x(t)}\Vert }{{\text{e}}_{\text{1}}{(t)}^T{P}_1{\text{e}}_{\text{1}}(t)}{\text{-A}}_{\text{i}}\text{ x(t)}-\frac{1}{2}\frac{{\text{e}}_{\text{1}}(t)\Vert \text{<mover accent="true"> <mi>f</mi> <mo>^</mo> </mover>(t)}\Vert {\Vert S_{\text{E}}\Vert }_{\max }\Vert \text{<mover accent="true"> <mi>f</mi> <mo>^</mo> </mover>(t)}\Vert }{{\text{e}}_{\text{1}}{(t)}^T{P}_1{\text{e}}_{\text{1}}(t)}+\text{S<mover accent="true"> <mi>f</mi> <mo>^</mo> </mover>(t)}\}\end{array}$$ E26

If

$${\text{e}}_{\text{1}}(t)=0$$ E27

$$\begin{array}{l}{u}_i(t)=Z^{-1}\left\{{\text{A}}_{\text{r}}\text{ <mover accent="true"> <mi>x</mi> <mo>¯</mo> </mover>(t)}+{\text{B}}_{\text{r}}\text{r(t)}-{\text{A}}_{\text{i}}\text{x(t)}+\text{S<mover accent="true"> <mi>f</mi> <mo>^</mo> </mover>(t)}\right\};\\ u_i(t)=B^T{Z}^{-1}\left\{{\text{A}}_{\text{r}}\text{ <mover accent="true"> <mi>x</mi> <mo>¯</mo> </mover>(t)}+{\text{B}}_{\text{r}}\text{r(t)}-{\text{A}}_{\text{i}}\text{x(t)}+\text{S<mover accent="true"> <mi>f</mi> <mo>^</mo> </mover>(t)}\right\}\end{array}$$ E28

where Z =B if B is an invertible square matrix or Z=BB ^T if B is not a square matrix, $$\Vert \cdot \Vert$$ denotes the l ₂ norm for vectors and l ₂ induced norm for matrices, $$\Vert S_{\text{E}}\Vert \le {\Vert S_{\text{E}}\Vert }_{\max }$$ , $$\Vert \Delta {\text{A}}_{\text{i}}\Vert \le {\Vert \Delta {\text{A}}_{\text{i}}\Vert }_{\max }$$ and $$\Vert D\Vert \le {\Vert D\Vert }_{\max }$$ , $$H\in {\kappa }^{\text{nxn}}$$ is a stable matrix to be designed and choosing S so that S=E and $$S_{\text{E}}=S^{T}S$$ , $$\text{D}={\text{B}}_{\text{ai}}^{\text{T}}{\text{B}}_{\text{ai}}$$ .

Theorem: If there exist symmetric and positive definite matrices P ₁₁ , P ₂₂ , some matrices K _i and L _i ,and matrices X _i , Y _i , such that the following LMIs are satisfied, then the TS fuzzy system (16) is asymptotically stabilizable via the TS fuzzy model based output-feedback controller (15),( 20) and(21)

$$A_i{}^T{\text{P}}_{\text{11}}+{\text{P}}_{\text{11}}{A}_i-{(Y_i{C}_i)}^T-Y_i{C}_i+{\text{P}}_{\text{11}}{\text{P}}_{\text{11}}-\delta I$$ E29

$${{\text{(X}}_{\text{i}}E)}^T+{\text{X}}_{\text{i}}E+P_{22}{P}_{22}-\delta I$$ E30

Proof. Before proceeding, we recall the following matrix inequality, which will be needed throughout the proof.

Lemma 2(Xie, 1996): Given constant matrices W and O appropriate dimensions for $$\forall \epsilon 0$$ , the following inequality holds:

$${\text{W}}^{\text{T}}\text{O}+O^{T}W\le \epsilon W^{T}W+{\epsilon }^{-1}{O}^{T}O$$ E31

Let

$${\text{e}}_{\text{1}}\text{(t)}=\text{x(t)-<mover accent="true"><mi>x</mi> <mo>¯</mo> </mover>(t)}$$ E32

$$\begin{array}{l}{\text{e}}_{\text{2}}\text{(t)}=\text{x(t)-<mover accent="true"> <mi>x</mi> <mo>^</mo> </mover>(t),}\\ \text{<mover accent="true"> <mi>f</mi> <mo>˜</mo> </mover>(t)}=\text{f(t)-<mover accent="true"> <mi>f</mi> <mo>^</mo> </mover>(t) }\end{array}$$ E33

The dynamic of e ₁ (t) is given by $${\text{<mover accent="true"> <mi>e</mi> <mo>˙</mo> </mover>}}_{\text{1}}\text{(t)}=\text{<mover accent="true"> <mi>e</mi> <mo>˙</mo> </mover>(t)-<mover accent="true"> <mover accent="true"> <mi>x</mi> <mo>¯</mo> </mover> <mo>˙</mo> </mover>(t)}$$

$${\text{<mover accent="true"> <mi>e</mi> <mo>˙</mo> </mover>}}_{\text{1}}\text{(t)}={\displaystyle \sum _{\text{i}=\text{1}}^{\text{p}}{{\displaystyle \mu }}_i\text{[}({\text{A}}_{\text{i}}+\Delta {\text{A}}_{\text{i}}\text{) x(t)}+B{{\displaystyle u}}_i{\text{(t) -A}}_{\text{r}}{\text{ <mover accent="true"> <mi>x</mi> <mo>¯</mo> </mover>(t)-B}}_{\text{r}}\text{r(t)]}}$$ E34

The dynamic of e ₂ (t) is expressed as follow:

$${\dot{e}}_2(t)={\displaystyle \sum _{i=1}^p{\mu }_i}{\text{[(A}}_{\text{i}}{\text{ -K}}_{\text{i}}{\text{C}}_{\text{i}}{\text{)e}}_{\text{2}}\text{(t)}-{\text{K}}_{\text{i}}\text{E<mover accent="true"> <mi>f</mi> <mo>˜</mo> </mover>(t)}+\Delta {\text{A}}_{\text{i}}\text{ x(t)]}$$ E35

The dynamics of the fault error estimation can be written: $$\text{<mover accent="true"> <mover accent="true"> <mi>f</mi> <mo>˜</mo> </mover> <mo>˙</mo> </mover>(t)}=\text{<mover accent="true"> <mi>f</mi> <mo>˙</mo> </mover>(t)-<mover accent="true"> <mover accent="true"> <mi>f</mi> <mo>^</mo> </mover> <mo>˙</mo> </mover>(t)}$$ . The assumption that the fault signal is constant over the time is restrictive, but in many practical situations where the faults are time-varying signals. So, we consider time-varying faults rather than constants faults; then the derivative of $$\text{<mover accent="true"> <mi>f</mi> <mo>˜</mo> </mover>(t)}$$ with respect to time is

$$\text{<mover accent="true"> <mover accent="true"> <mi>f</mi> <mo>˜</mo> </mover> <mo>˙</mo> </mover>(t)}=\text{<mover accent="true"> <mi>f</mi> <mo>˙</mo> </mover>(t)-<mover accent="true"> <mover accent="true"> <mi>f</mi> <mo>^</mo> </mover> <mo>˙</mo> </mover>(t)}=\text{<mover accent="true"> <mi>f</mi> <mo>˙</mo> </mover>(t)-}{\displaystyle \sum _{i=1}^p{\mu }_i}{\text{[L}}_{\text{i}}{\text{C}}_{\text{i}}{\text{e}}_{\text{2}}\text{(t)}+{\text{L}}_{\text{i}}\text{E<mover accent="true"> <mi>f</mi> <mo>˜</mo> </mover>(t)]}$$ E36

From (29) and (30), one can obtain:

$$\dot{\phi }={\text{A}}_o\phi +B_o\text{<mover accent="true"> <mi>f</mi> <mo>˙</mo> </mover>(t)}+{\text{B}}_{\text{a}}x(t)$$ E37

$$\begin{array}{l}with,\phi =\left[\begin{array}{l}{e}_2(t)\\ \text{<mover accent="true"> <mi>f</mi> <mo>˜</mo> </mover>(t)}\end{array}\right],B_o=\left[\begin{array}{l}0\\ \text{I}\end{array}\right],\\ A_o={\displaystyle \sum _{i=1}^p{\mu }_i}{A}_{oi},B_a={\displaystyle \sum _{i=1}^p{\mu }_i}{B}_{ai},where\\ {\text{A}}_{\text{oi}}=\left[\begin{array}{l}{A}_i-K_i{C}_i-K_{i}E\\ -L_{i}C{}_i-L_{i}E\end{array}\right],{\text{B}}_{\text{ai}}=\left[\begin{array}{l}\Delta A_i\\ 0\end{array}\right]\end{array}$$ E38

Consider the Lyapunov function candidate

$$V(e_1(t),\phi (t))=\frac{1}{2}{e}_1{\text{(t)}}^{\text{T}}{\text{P}}_{\text{1}}{e}_1\text{(t)}+\frac{1}{2}\phi {\text{(t)}}^{\text{T}}{\text{P}}_{\text{2}}\phi \text{(t)}$$ E39

where P ₁ and P ₂ are time-invariant, symmetric and positive definite matrices. Let

$$\begin{array}{l}{V}_1(e_1)=\frac{1}{2}{e}_1{\text{(t)}}^{\text{T}}{\text{P}}_{\text{1}}{e}_1\text{(t),}\\ V_2(\phi )=\frac{1}{2}\phi {\text{(t)}}^{\text{T}}{\text{P}}_{\text{2}}\phi \text{(t)}\end{array}$$ E40

The time derivative of V ₁ (e ₁ (t))is

$${\dot{V}}_1(e_1(t))=\frac{1}{2}{\dot{e}}_1{\text{(t)}}^{\text{T}}{\text{P}}_{\text{1}}{e}_1\text{(t)}+\frac{1}{2}{e}_1{\text{(t)}}^{\text{T}}{\text{P}}_{\text{1}}{\dot{e}}_1\text{(t)}$$ E41

By substituting (28) into (34) yields

$$\begin{array}{l}{\dot{V}}_1=\frac{1}{2}\{{\displaystyle \sum _{\text{i}=\text{1}}^{\text{p}}{{\displaystyle \mu }}_i{\text{[(A}}_{\text{i}}+\Delta {\text{A}}_{\text{i}}\text{ ) x(t)}+B{{\displaystyle u}}_i{\text{(t) -A}}_{\text{r}}\text{ <mover accent="true"> <mi>x</mi> <mo>¯</mo> </mover>(t)}}-{\text{B}}_{\text{r}}\text{r(t)]}\}{\text{P}}_{\text{1}}{e}_1\text{(t)}\\ +\frac{1}{2}{e}_1{\text{(t)}}^{\text{T}}{\text{P}}_{\text{1}}\left\{{\displaystyle \sum _{\text{i}=\text{1}}^{\text{p}}{{\displaystyle \mu }}_i{\text{[(A}}_{\text{i}}+\Delta {\text{A}}_{\text{i}}\text{ ) x(t)}+B{u}_i{\text{(t)-A}}_{\text{r}}\text{ <mover accent="true"> <mi>x</mi> <mo>¯</mo> </mover>(t)}-{\text{B}}_{\text{r}}\text{r(t)]}}\right\}\end{array}$$ E42

we design u _i (t), i=1,2,…,p as follows,

• When B is an invertible square matrix, the control law is given by

• When B is not a square matrix, the control law is given by

$$u_i(t)=B^T{Z}^{-1}{Z}_{ui}$$ E44

where Z _ui is given from (22). A block diagram of the closed-loop system is shown in Fig.4. It is assumed that Z ^-1 exists (Z is nonsingular). In the latter part of this section, we shall provide a way to check if the assumption is valid. From (35), (36) or (37) and assuming that e ₁ (t)≠0 and using Lemma 2, one obtain

$$\begin{array}{l}{\dot{V}}_1=\frac{1}{2}{e}_1{\text{(t)}}^{\text{T}}{\text{(H}}^{\text{T}}{\text{P}}_{\text{1}}+{\text{P}}_{\text{1}}H+{\text{P}}_{\text{1}}{\text{P}}_{\text{1}})e_1\text{(t)}\\ +\frac{1}{2}{\displaystyle \sum _{i=1}^p{\mu }_i}\Vert {\text{e}}_{\text{1}}(t)\Vert \Vert P_1\Vert (\Vert \Delta {\text{A}}_{\text{i}}\Vert \text{-}{\Vert \Delta {\text{A}}_{\text{i}}\Vert }_{\max }\text{) }\Vert \text{x(t)}\Vert \\ +\frac{1}{2}(\Vert \text{<mover accent="true"> <mi>f</mi> <mo>^</mo> </mover>(t)}\Vert (\Vert S_{\text{E}}\Vert -{\Vert S_{\text{E}}\Vert }_{\max }\text{)}\Vert \text{<mover accent="true"> <mi>f</mi> <mo>^</mo> </mover>(t)}\Vert -\Vert \text{x(t)}\Vert {\Vert D\Vert }_{\max }\Vert \text{x(t)}\Vert )\end{array}$$ E45

The time derivative of V ₂ (ϕ (t)) is

$${\dot{V}}_2(\phi (t))=\frac{1}{2}\dot{\phi }{\text{(t)}}^{\text{T}}{\text{P}}_{\text{2}}\phi \text{(t)}+\frac{1}{2}\phi {\text{(t)}}^{\text{T}}{\text{P}}_{\text{2}}\dot{\phi }\text{(t)}$$ E46

By substituting (31) into (39) yields

$$\begin{array}{l}{\dot{V}}_2(\phi (t))=\frac{1}{2}{\displaystyle \sum _{i=1}^p{\mu }_i\phi {\text{(t)}}^{\text{T}}({{\displaystyle A}}_{oi}^T{\text{P}}_{\text{2}}+{\text{P}}_{\text{2}}{A}_{oi}})\phi \text{(t)}\\ +\frac{1}{2}{\displaystyle \sum _{i=1}^p{\mu }_i}[{\text{<mover accent="true"> <mi>f</mi> <mo>˙</mo> </mover>(t)}}^{\text{T}}{B}_o^{\text{T}}{\text{P}}_{\text{2}}\phi \text{(t)}+\phi {\text{(t)}}^{\text{T}}{\text{P}}_{\text{2}}{B}_o\text{<mover accent="true"> <mi>f</mi> <mo>˙</mo> </mover>(t)}]\\ +\frac{1}{2}{\displaystyle \sum _{i=1}^p{\mu }_i}[{\text{x(t)}}^{\text{T}}{B}_{ai}^T{\text{P}}_{\text{2}}\phi \text{(t)}+\phi {\text{(t)}}^{\text{T}}{\text{P}}_{\text{2}}{B}_{ai}\text{x(t)}]\end{array}$$ E47

Using Lemma 2 and the definition ( Billy, E. (2011 )) $${\text{<mover accent="true"> <mi>f</mi> <mo>˙</mo> </mover>(t)}}^{\text{T}}{B}_o^{\text{T}}B\text{<mover accent="true"> <mi>f</mi> <mo>˙</mo> </mover>(t)}={\text{f}}_1^2{\lambda }_{\max }^{}(B_o^{\text{T}}{B}_o\text{)}$$ , one obtain

$${\dot{V}}_2(\phi (t))=\frac{1}{2}{\displaystyle \sum _{i=1}^p{\mu }_i\phi {\text{(t)}}^{\text{T}}({{\displaystyle A}}_{oi}^T{\text{P}}_{\text{2}}+{\text{P}}_{\text{2}}{A}_{oi}+{\text{P}}_{\text{2}}{\text{P}}_{\text{2}}})\phi \text{(t)}+\frac{1}{2}{\text{f}}_1^2{\lambda }_{\max }^{}(B_o^{\text{T}}{B}_o\text{)}+\frac{1}{2}{\text{x(t)}}^{\text{T}}D\text{x(t)}$$ E48

where $${\lambda }_{\max }^{}(•\text{)}$$ denotes the largest eigen value. Combining (38) with (41), the time derivative of V can be expressed as

$$\begin{array}{l}\dot{V}(e_1(t),\phi (t))\le -\frac{1}{2}{e}_1{\text{(t)}}^{\text{T}}{Q}_1{e}_1\text{(t)}-\phi {\text{(t)}}^{\text{T}}{Q}_2\phi \text{(t)}\\ +\frac{1}{2}\Vert \text{<mover accent="true"> <mi>f</mi> <mo>^</mo> </mover>(t)}\Vert (\Vert S_{\text{E}}\Vert -{\Vert S_{\text{E}}\Vert }_{\max })\Vert \text{<mover accent="true"> <mi>f</mi> <mo>^</mo> </mover>(t)}\Vert \\ +{\displaystyle \sum _{i=1}^p{\mu }_i}\Vert {\text{e}}_{\text{1}}(t)\Vert \Vert P_1\Vert (\Vert \Delta {\text{A}}_{\text{i}}\Vert \text{-}{\Vert \Delta {\text{A}}_{\text{i}}\Vert }_{\max }\text{) }\Vert \text{x(t)}\Vert +\Vert \text{x(t)}\Vert (\Vert D\Vert -{\Vert D\Vert }_{\max })\Vert \text{x(t)}\Vert \end{array}$$ E49

where $$Q_2=-({{\displaystyle A}}_{oi}^T{\text{P}}_{\text{2}}+{\text{P}}_{\text{2}}{A}_{oi}+{\text{P}}_{\text{2}}{\text{P}}_{\text{2}}+\delta )$$ , $${\text{Q}}_{\text{1}}={\text{-(H}}^{\text{T}}{\text{P}}_{\text{1}}+{\text{P}}_{\text{1}}\text{H}+{\text{P}}_{\text{1}}{\text{P}}_{\text{1}})$$ are a symmetric positive definite matrix, where $$\delta =0.5{\text{f}}_1^2{\lambda }_{\max }(B_o^{\text{T}}{B}_o\text{)}$$ , as $$\Vert S_E\Vert \text{-}{\Vert S_E\Vert }_{\max }\le 0$$ , $$\Vert \Delta {\text{A}}_{\text{i}}\Vert \le {\Vert \Delta {\text{A}}_{\text{i}}\Vert }_{\max }$$ , $$\Vert D\Vert \le {\Vert D\Vert }_{\max }$$ , from (42), we have

$$\dot{V}(e_1(t),\phi (t))\le -\frac{1}{2}{e}_1{\text{(t)}}^{\text{T}}{Q}_1{e}_1\text{(t)}-\phi {\text{(t)}}^{\text{T}}{Q}_2\phi \text{(t)}$$ E50

e ₁ and ϕ converges to zero if $$\dot{V}0$$ . $$\dot{V}0$$ if there exists a common positive definite matrix P ₁ and P ₂ such that

$${\text{H}}^{\text{T}}{\text{P}}_{\text{1}}+{\text{P}}_{\text{1}}\text{H}+{\text{P}}_{\text{1}}{\text{P}}_{\text{1}}0$$ E51

$$\begin{array}{l}{{\displaystyle \text{A}}}_{\text{oi}}^{\text{T}}{\text{P}}_{\text{2}}+{\text{P}}_{\text{2}}{A}_{oi}+{\text{P}}_{\text{2}}{\text{P}}_{\text{2}}-\delta I\\ i=1,2,\dots ,p\end{array}$$ E52

From (43), (44) and (45)

$$\dot{V}\le -\frac{1}{2}{e}_1{\text{(t)}}^{\text{T}}{Q}_1{e}_1\text{(t)}-\phi {\text{(t)}}^{\text{T}}{Q}_2\phi \text{(t)}\le \text{0}$$ E53

If the time derivative of (32) is negative uniformly for all $$e_1(t),\phi (t)$$ and for all $$t\ge 0$$ except at $$e_1(t)=0,\phi (t)=0$$ then the controlled fuzzy system (16) is asymptotically stable about its zero equilibrium. Since each sum of the equation in (46) is negative definite, respectively, then the controlled TS fuzzy system is asymptotically stable. According to the (45), the most important step in designing the fuzzy observer based fuzzy controller is the solution of (45) for a common P ₂ = P ₂ ^T , a suitable set of observer gains K _i and L _i (i =1,2,…,p). Equation (45) forms a set of bilinear matrix inequalities (BMI's). The BMI in (45) should be transformed into pure LMI as follows:

1. For the convenience of design, assume P ₂ =diag(P ₁₁ ,P ₂₂ )

2. This choice is suitable for simplifying the design of fuzzy observer. Define M as in (47) and apply congruence transformation (i.e. multiply both side of inequality (45) by M (Tuan et al., 2001).

$$M=\left[\begin{array}{l}{M}_{11}0\\ 0I\end{array}\right]=\left[\begin{array}{l}{P}_{11}^{-1}0\\ 0I\end{array}\right]$$ E54

where

$$M_{11}=P_{11}^{-1}$$ E55

1. With Y _i =P ₁₁ K _i and X _i =-P ₂₂ L _i , apply the change of variables. Since four parameters P ₁₁ , P ₂₂ , K _i and L _i should be determined from (45) after applying the above mentioned procedures which results in the LMI's conditions (24) and (25).

The inequalities in (24) and (25) are linear matrix inequality feasibility problems (LMIP's) in P ₁₁ , P ₂₂ , Y _j and X _i which can be solved very efficiently by the convex optimization technique such as interior point algorithm (Boyd et al., 1994). Software packages such as LMI optimization toolbox in Matlab (Gahinet et al., 1995) have been developed for this purpose and can be employed to easily solve the LMIP. By solving (24) and (25) the observer gain (K _i and L _i ) can be easily determined.

In the following, we shall derive a sufficient condition to check the existence of Z ^-1 . From (17), and considering the following dynamics system,

$$\dot{\theta }(t)=Z\theta (t)={\displaystyle \sum _{\text{i}=\text{1}}^{\text{p}}{{\displaystyle \mu }}_i}{Z}_i\theta (t)$$ E56

Consider the following Lyapunov function

$$V_z=\theta {\text{(t)}}^{\text{T}}\text{P}\theta \text{(t)}$$ E57

where $$P\in {\kappa }^{\text{nxn}}$$ is a symmetric positive definite matrix. The time derivative of $$V_z$$ is

$${\dot{V}}_z=\dot{\theta }{\text{(t)}}^{\text{T}}\text{P}\theta \text{(t)}+\theta {\text{(t)}}^{\text{T}}\text{P}\dot{\theta }\text{(t)}$$ E58

By substituting (48) into (50) yields

$$\dot{V}=-{\displaystyle \sum _{\text{i}=\text{1}}^{\text{p}}{{\displaystyle \mu }}_i}\theta {\text{(t)}}^{\text{T}}{Q}_{zi}\theta \text{(t)}$$ E59

where $${\text{Q}}_{\text{zi}}=\text{-(}{Z}_i{}^T\text{P}+\text{P}{Z}_i)$$ . If $$Z_i{}^T\text{P}+\text{P}{Z}_{i}0\forall i$$ , then from (51), we have,

$${\dot{V}}_z=-\frac{1}{2}{\displaystyle \sum _{\text{i}=\text{1}}^{\text{p}}{{\displaystyle \mu }}_i}\theta {\text{(t)}}^{\text{T}}{Q}_{zi}\theta \text{(t)}\le \text{0}$$ E60

From (52), the nonlinear system of (48) is then asymptotically stable and Z ^-1 exists if there exist $${\text{Z}}_{\text{i}}^{\text{T}}\text{P}+\text{P}{Z}_{i}0\forall i$$

Figure 4.

Block diagram of RFFTC scheme

5. TS Fuzzy description and control structure

5.1. Control structure

Fig. 5 depicts the input and output relationship of the wind-battery system from the control point of view. The control inputs are the excitation field voltage (E _fd ) of the SG and the direct-current set point (I _ref ) of the converter. The measurements are the voltage amplitude (V _b ) and the frequency (f) of the AC bus. The wind speed (ν) and the load (ν ₁ ) are considered to be disturbances. The wind turbine generator and the battery-converter unit run in parallel, serving the load. From the control point of view, this is a coupled 2×2 multi-input-multi-output nonlinear system.

Figure 5.

The HWDSS control system

5.2. TS Fuzzy WECS description

To design the fuzzy fault tolerant controller and the fuzzy observer, we must have a fuzzy model that represents the dynamics of the nonlinear plant. The TS fuzzy model that approximates the dynamics of the nonlinear HWDSS plant (7) can be represented by the following four-rule fuzzy model. Referring to (10) the fuzzy plant model given by:

$$\begin{array}{l}\stackrel{.}{x}(t)={\displaystyle \sum _{i=1}^4{h}_i}\text{[}({\text{A}}_{\text{i}}+\Delta {\text{A}}_{\text{i}}\text{) x(t)}+{{\displaystyle B}}_i\text{u(t)]}\\ y(t)={\displaystyle \sum _{\text{i}=\text{1}}^{\text{4}}{{\displaystyle \mu }}_i\text{[}{C}_{\text{i}}\text{x(t)}+{\text{E}}_{\text{i}}\text{f(t)] }}\end{array}$$ E61

where $$\text{x}(t)\in {\kappa }^{\text{2x1}}$$ , $$\text{u}(t)\in {\kappa }^{\text{2x1}}$$ are the state vectors and the control input, respectively. For each sub-space, different model ( $$i=1,2,3,4$$ ) and ( $$p=4$$ ) is applied. The degree of membership function for states V _b and ω _s is depicted in Fig.6. Where

$$\begin{array}{l}{A}_g={{\displaystyle \left[\begin{array}{l}\text{-0}\text{.006}\text{1}\text{.213}\\ 0\text{-0}\text{.002}\end{array}\right]}}_{},A_j={{\displaystyle \left[\begin{array}{l}\text{-0}\text{.0063 }\text{1}\text{.207}\\ 0\text{-0}\text{.002}\end{array}\right]}}_{},B_g=\left[\begin{array}{l}1\text{-0}\text{.5808}\\ 0\text{-0}\text{.5808}\end{array}\right],\\ B_j=\left[\begin{array}{l}1\text{-0}\text{.5979}\\ 0\text{-0}\text{.5979}\end{array}\right],E_g=E_j=E=\left[\begin{array}{l}101\\ 0.10.01\end{array}\right],g=1,2andj=3,4\end{array}$$ E62

$$\Delta {\text{A}}_{\text{i}}$$ represent the system parameters uncertainties but bounded, the elements of $$\Delta {\text{A}}_{\text{i}}$$ randomly achieve the values within 40% of their nominal values corresponding to $${\text{A}}_{\text{i}}$$ , $$\Delta {\text{B}}_{\text{i}}=0$$ (i=1,2,3,4), and the faults f(t) are modeled as follow:

$${\text{f}}_{\text{1}}\text{(t) }=\{\begin{array}{l}\text{0 }\text{ t}\text{2}0.7\text{5}\text{sec }\\ \text{5}\text{.9 sin(}\pi \text{t) t}\ge \text{2}0.7\text{5}\text{sec }\end{array},{\text{f}}_{\text{2}}\text{(t) }=\{\begin{array}{l}\text{0 t}\le \text{2}0.7\text{5}\text{sec }\\ \text{1 }\text{t}\ge \text{2}0.7\text{5}\text{sec }\end{array}$$ E63

where f ₁ (t) is the bus voltage sensor fault and f ₂ (t) is the generator speed sensor fault.

Figure 6.

Membership functions of states ω _s and V _b

6. Procedures for finding the robust fuzzy fault tolerant controller and fuzzy observer

According to the analysis above, the procedure for finding the proposed fuzzy fault tolerant controller and the fuzzy observer summarized as follows.

1. Obtain the mathematical model of the HWDSS to be controlled.

2. Obtain the fuzzy plant model for the system stated in step (1) by means of a fuzzy modeling method.

3. Check if there exists Z ^-1 by finding P according to Lemma 1. If P cannot be found, the design fails. P can be found by using some existing LMI tools.

4. Choose a stable reference model.

5. Solve LMIs (24) and (25) to obtain X _i , Y _i , P ₁₁ , P ₂₂ , K _i and L _i thus (K _i =P ₁₁ ^-1 Yi and L _i =- P ₂₂ ^-1 X _i ).

6. Construct fuzzy observer (12) according to the theorem and fuzzy controller (14) controller according to the Lemma 1.

7. Simulation studies

The simulations are performed on a simulation model of hybrid wind-diesel storage system (7). The proposed Fuzzy Fault Tolerant controller for the HWDSS is tested for two cases. The proposed controller is tested for random variation of wind speed signal as shown in Fig.7 to prove the effectiveness of the proposed algorithm. The reference input (r(t)=ω _t(opt) =νλ _opt /R) is applied to the reference model and the controller to obtain the maximum power coefficient from the wind energy.

7.1. System responses of the fuzzy control system without and with parameter uncertainties

Fig. 8 shows the HWDSS responses of the fuzzy control system without (solid lines) and with parameter uncertainties (dash lines), and the reference model (dotted lines) under r(t).

Figure 7.

Wind speed

Figure 8.

Responses of bus voltage (V _b ) and rotor speed (ω _s ) of the fuzzy control system without (solid line) and with parameter uncertainties (dash line), and the reference model (dotted line) under r(t)

7.2. System responses of the fuzzy control system with sensor faults and with parameter uncertainties

The Fig.9 (top) shows the time evolution of the sensor fault $$\text{f(t)}$$ (solid lines) and its estimate $$\text{<mover accent="true"> <mi>f</mi> <mo>^</mo> </mover>(t)}$$ (dash lines) based on (54) while the bottom part shows the fault estimation errors $$\text{<mover accent="true"> <mi>f</mi> <mo>˜</mo> </mover>(t)}$$ . The response of the HWDSS states (solid lines), the states of the observer (dash lines) and reference model states (dotted lines) are given in Fig.10. The state estimation errors ( $$V_b-{\widehat{V}}_b$$ , $${\omega }_s-{\widehat{\omega }}_s$$ ) are shown in the top of Fig.11, while the bottom part shows the state tracking errors( $$V_b-{\overline{V}}_b$$ , $${\omega }_s-{\overline{\omega }}_s$$ ). As the wind speed varying as the random variation, the produced power curve takes almost the wind speed curve as shown in Fig. 12, but there is only spike when the fault is detected at 20.75 sec.

Figure 9.

Faults and their estimations (bus voltage sensor fault f ₁ (t) and its estimate and generator speed sensor fault f ₂ (t) and its estimate)(top) Fault estimation errors (bottom)

Figure 10.

Responses of bus voltage (V _b ) and rotor speed (ω _s ) of the fuzzy control system (solid line), observer (dash line) and the refence model (dotted line) with parameter uncertainties and sensor faults based on (54) under the same refence input r(t)

It can be seen from the simulation results that the states of the HWDSS system follow those of the reference model in the presence bounded parametric uncertainties and sensor faults. Fig. 8 shows that the responses of the fuzzy control system with parameter uncertainties are better than that of the fuzzy control system without parameter uncertainties. This is because an additional control signals, i.e., $$\frac{{\text{e}}_{\text{1}}(t)\Vert {\text{e}}_{\text{1}}(t)\Vert \Vert P_1\Vert {\Vert \Delta {\text{A}}_{\text{i}}\Vert }_{\max }\Vert \text{x(t)}\Vert }{{\text{e}}_{\text{1}}{(t)}^T{P}_1{\text{e}}_{\text{1}}(t)},\frac{{\text{e}}_{\text{1}}(t)\Vert \text{x(t)}\Vert {\Vert D\Vert }_{\max }\Vert \text{x(t)}\Vert }{{\text{e}}_{\text{1}}{(t)}^T{P}_1{\text{e}}_{\text{1}}(t)}$$ are used, the reason can also be seen from (42), i.e., $${\displaystyle \sum _{i=1}^p{\mu }_i}\Vert {\text{e}}_{\text{1}}(t)\Vert \Vert P_1\Vert (\Vert \Delta {\text{A}}_{\text{i}}\Vert \text{-}{\Vert \Delta {\text{A}}_{\text{i}}\Vert }_{\max }\text{) }\Vert \text{x(t)}\Vert$$ $$+\Vert \text{x(t)}\Vert (\Vert D\Vert -{\Vert D\Vert }_{\max })\Vert \text{x(t)}\Vert$$ makes error e(t) approach zero at a faster rate. Figs. 10 shows there is spike when the fault is detected at 20.75 sec and then the HWDSS trajectory follows the trajectory of the reference model, this is because an additional control signals, $${\text{e}}_{\text{1}}(t)\Vert \text{<mover accent="true"> <mi>f</mi> <mo>^</mo> </mover>(t)}\Vert {\Vert S_{\text{E}}\Vert }_{\max }\Vert \text{<mover accent="true"> <mi>f</mi> <mo>^</mo> </mover>(t)}\Vert /{\text{e}}_{\text{1}}{(t)}^T{P}_1{\text{e}}_{\text{1}}(t)$$ .

Figure 11.

State estimation errors ( $$V_b-{\widehat{V}}_b$$ , $${\omega }_s-{\widehat{\omega }}_s$$ )(top) State tracking errors ( $$V_b-{\overline{V}}_b$$ , $${\omega }_s-{\overline{\omega }}_s$$ ) (bottom)

Figure 12.

Per unit wind turbine produced power

In summery results, we can be seen that the system trajectory follows the trajectory of the reference model which represents the trajectory of the HWDSS in the fault free situation. Thus, the TS fuzzy model based controller through fuzzy observer is robust against norm-bounded parametric uncertainties and sensor faults. Comparing the results of the proposed algorithm, with that given in the previous algorithms, we can be seen that the proposed controller has the following advantages:

1. It can control the plant well over a wide range of sensor faults compared with (Wei et al., 2010 ; Odgaard et al., 2009; Gaillard et al., 2007).

2. Is stable over a wide range of uncertainty up to 40% compared with (Uhlen et al., 1994).

3. The generated power is increased up to 45% compared with (Chen & Hu, 2003; Kamal et al., 2010).

4. The algorithm is more robust in the presence of high nonlinearity.

5. Bus voltage is nearly constant and voltage ripple is reduced to 25% compared with (Chedid et al., 2000; Kamal et al., 2010).

8. Chapter conclusion

The stability analysis and design of nonlinear HWDSS control systems have been discussed. An improved stability criterion has been derived. In this chapter, we have developed a new robust fuzzy fault tolerant controller to control a HWDSS, while taking into account sensor fault(s) and parametric uncertainties in the aerodynamic model under the conditions that the state variables are unavailable for measurement as well as enabling the system to capture as much wind power as possible. A reference model is used and the proposed control is then designed for guaranteeing the convergence of the states of the HWDSS to the states of a reference model even if sensor fault(s) occurs and with parametric uncertainties. The basic approach is based on the rigorous Lyapunov stability theory and the basic tool is LMI. Some sufficient conditions for robust stabilization of the TS fuzzy model are formulated in the LMIs format. The closed-loop system will behave like a user-defined reference model in the presence of bounded sensor faults and parameter uncertainties. A simulation on HWDSS has been given to show the design procedure and the merits of the proposed fuzzy fault tolerant controller.