A Fuzzy Approach for Sensor Fault-Tolerant Control of ...

EUSFLAT-LFA 2011

July 2011

Aix-les-Bains, France

A Fuzzy Approach for Sensor Fault-Tolerant Control of wind energy conversion Systems Elkhatib KAMAL1 Abdel AITOUCHE2 Mireille BAYART1 LAGIS FRE CNRS 3033, Lille University,Nord, France e-mail: [email protected] 2 LAGIS FRE CNRS 3033, Lille University,Nord, HEI, France e-mail: [email protected] 1

Abstract despite the presence of bounded sensor faults. Based on the derivation, we obtain sufficient conditions, expressed in LMI terms, for the existence of robust fuzzy controllers for TS fuzzy model with sensor faults. The state feedback gains of a robust fuzzy controller and the gains of the fuzzy observer can also be directly obtained from the LMI solutions. This paper is organized as follows: Section 2 describes the fuzzy plant model, fuzzy observer and reference model. The Proposed FTC Fuzzy controller and the condition for stability are presented in section 3. Section 4 presents description of WECS model and TS fuzzy description. In Section 5 simulation results illustrate the effectiveness of the proposed control method for wind systems. A conclusion is drawn in section 6.

This paper presents the robust fuzzy fault tolerant control (FFTC) for nonlinear wind energy conversion systems (WECS) in the presence of bounded sensor faults and the state variable unavailable for measurement based on Takagi-Sugeno (TS) fuzzy model. Sufficient conditions are derived for robust stabilization in the sense of Lyapunov asymptotic stability and are formulated in the format of linear matrix inequalities (LMIs). The closed-loop system will follow those of a userdefined stable reference model in the presence of bounded sensor faults. The effectiveness of the proposed fuzzy fault tolerant controller and fuzzy observer design methodology is finally demonstrated through simulations on the WECS. Keywords: Takagi-Sugeno observer; LMI; WECS; FTC; Fuzzy Proportional Integral Observer.

2. TS fuzzy model , reference model and fuzzy observer

1. Introduction 2.1. TS fuzzy plant model with sensor faults

Stability is one of the most important problems in the analysis and synthesis of control systems. Recently, the issue of stability of fuzzy control systems in nonlinear stability frameworks has been considered extensively [1], [2]. Ref. [2] presented a design method for the stabilization of a class of nonlinear systems as described by TS fuzzy model. In order to design a fuzzy controller, they used the concept of the so-called parallel distributed compensation (PDC) and linear matrix inequality (LMI). Ref. [3] also presented stability conditions satisfying decay rate for TS fuzzy model. Since faults are frequently a source of instability and encountered in various engineering systems, the issue of a robust fuzzy controller design for faulty nonlinear systems has received considerable interest [4],[5]. Ref. [4] derived robust stability conditions for a fuzzy system with sensor faults. Ref. [5] also presented active fault-tolerant control for nonlinear systems with sensor faults and a method for designing robust controllers to stabilize sensor fault nonlinear systems. More recently, in [6]-[8] the fault tolerant control strategy for wind energy conversion systems has been well developed and extensively applied to efficiently deal with the problems of robust stabilization and disturbance rejection. This paper is dedicated to the design of a fuzzy fault tolerant control strategy for nonlinear systems with sensor faults described by TS models. This approach is an extension of the work proposed in [4], [5]. In designing a fuzzy FTC control system, the nonlinear systems are represented by TS fuzzy model with sensor faults. PDC is employed to design the fuzzy controllers from TS fuzzy model. A sufficient condition is derived so that the closed-loop system is asymptotically stable and will follow those of a user-defined stable reference model

© 2011. The authors - Published by Atlantis Press

The TS fuzzy model is described by fuzzy IF-THEN rules, which represent local linear input-output relationships of nonlinear systems [9]. The ith rule of the TS fuzzy model with sensor fault is in the following form. Plant Rule i: q1(t) is Ni1 AND … AND qψ(t) is Niψ Then x& (t ) = A i x(t) + B i U(t) , y (t ) = Ci x(t ) + E i f(t)

(1)

i

where N Ω is a fuzzy set of rule i, Ω = 1,2,…,ψ, i=1,2,…,p, x (t ) ∈ κ nx1 is the state vector, U(t ) ∈ κ nx1 is the input vector, y(t ) ∈ κ

f (t ) ∈ κ

rx1

mx1

is the output vector,

represents the fault which is assumed to be

bounded, A ∈ κ i

nxn

and B ∈ κ i

nxn

, C ∈κ i

gxn

,

E

i

are system matrix, input matrix, output matrix and fault matrix, respectively, which are assumed to be known. It is supposed that the matrix Ei is full column rank, i.e. rank(Ei)= r. It is assumed that the derivative of f(t) w. r.t to time is norm bounded, i.e. f& (t ) ≤ d and 1

0 ≤ d1 < ∞ , p is the number of IF-THEN rules, and q1(t),…, qψ(t) are assumed measurable variables and do not depend on the sensor faults [10]. The defuzzified output of (2) subject to sensor faults is represented as follows [4]: p

x& (t ) = ∑ µ i (q(t))[ Ai x(t) + Bi U(t)] i =1 p

y (t ) = ∑ µ i (q(t))[Ci x(t) + E i f(t)] i =1

791

, (2)

Where

3. Proposed FFTC algorithm design ψ

hi ( q(t )) = Π Nαi ( q(t )) , µi (q(t)) = α =1

h i ( q(t ))

In this section, we present LMI-based solutions to the fuzzy FTC controller synthesis problems for nonlinear systems with sensor faults described by the TS fuzzy model. The proposed fuzzy FTC controller can be designed such that the states of the closed-loop system will follow those of a user-defined stable reference model (1) despite the presence of sensor faults.

p

∑ h i ( q(t ))

i =1

Some basic properties of p

0 ≤ µi (q(t)) ≤ 1,

∑ µ i (q(t)) = 1 ∀i = 1,2,...., p

(3)

i =1

3.1. PDC technique

2.2. Reference model

The concept of PDC in [2] is utilized to design fuzzy controllers to stabilize fuzzy system (2). The idea of PDC is to associate a compensator for each rule of the fuzzy model. The resulting overall fuzzy controller is a fuzzy blending of each individual linear controller. The fuzzy controller shares the same fuzzy sets with the fuzzy system (2).

A reference model is a stable linear system without faults given by [9], x& ( t ) = A r x (t) + B r r(t) y (t ) = Cr x (t ) Where x (t ) ∈ κ el,

r (t ) ∈ κ

A ∈κ

nxn

∈κ

nxn

r

B

r

nx1

nx1

is

(4)

is the state vector of reference modthe

bounded

reference

3.2. Proposed fuzzy controller

input,

Definition 2: If the pairs (Ai,Bi), i= 1, 2,…, p, are controllable, the fuzzy system (1) is called locally controllable[11]. For the fuzzy controller design, it is assumed that the fuzzy system (1) is locally controllable. First, the local state feedback controllers are designed as follows, based on the pairs (Ai,Bi). Using PDC the ith rule of the fuzzy controller which is the following format, Controller Rule i: q1(t) is Ni1 AND … AND qψ(t) is Niψ Then u(t)=ui(t) (7)

is the constant stable system matrix, is the constant input matrix, C ∈ κ

mxn

r

the constant output matrix. y(t ) ∈ κ output.

mx1

is

is the reference

2.3. Fuzzy Proportional Integral Observer (FPIO) design

where u (t ) ∈ κ

Definition 1: If the pairs (Ai,Ci), i= 1, 2,…, p, are observable, the fuzzy system (1) is called locally observable[11]. For the fuzzy observer design, it is assumed that the fuzzy system (1) is locally observable. First, the local state observers are designed as follows, based on the triplets (Ai,Bi,,Ci). In order to detect and estimate faults, the following fault estimation fuzzy state observer for TS fuzzy model with sensor faults (1) is formulated as follows [4],[5]: Observer Rule i: IF q1(t) is Ni1AND … AND qψ(t) is Niψ Then xˆ& (t ) = A xˆ (t) + B u(t) + K (y(t) - yˆ(t)) ,

ler that will be defined in the next sub-section. The global output of the fuzzy controller is given by

i

i

p

3.3.

(8)

Design of the proposed FFTC controller

We design the control law ui(t) for i=1,2,…,p, such that closed-loop system behaves like the stable reference model. From (2), (3) and (8), writing µ i (q(t)) as µ i , we have, p

x& (t ) = ∑ µ i [ Ai x(t) + Bi u i (t) ] i =1 p

y (t ) = ∑ µ i [C i x(t) + E i f(t) ] i =1

(5)

(9)

we use the property

Where Ki is the proportional observer gain for the ith observer rule and Li are their integral gains to be determined. yˆ (t ) is the final output of the fuzzy observer. ~ y (t ) is the estimation error. The defuzzified output of

p

p

p

p

p

∑ µ i = ∑ ∑ µ iµ j = 1 , B = ∑ µ i B , E = ∑ µ i E (10) i i i=1 j = 1 i =1 i =1 i =1

Note that B and E are known. Also from (3), (4) and (8), we have

(5) subject to sensor faults is represented as follows:

p

x&ˆ (t ) = ∑ µ i (q(t))[ Ai xˆ(t) + Bi u i (t) + K i ~y(t)]

p

x&ˆ (t ) = ∑ µ i (q(t))[ Ai xˆ(t) + Bi U(t) + K i ~y(t)]

i =1 p

i =1

yˆ (t ) = ∑ µ i (q(t))[C i xˆ(t) + E i fˆ(t)]

p & fˆ(t) = ∑ µi Li ~y

(11)

i =1

i =1 p

yˆ (t ) = ∑ µ i (q(t))[Ci xˆ(t) + E i fˆ(t)]

is the output of the ith rule control-

U (t ) = ∑ µ i (q(t)) u i (t) i =1

i

& fˆ(t) = L (y - yˆ) = L ~y i i ˆy (t ) = C xˆ (t ) + E fˆ(t) i=1, 2,…,p i i

nx1

i

let (6)

i =1

e (t) = x(t) - x(t)

(12)

~ e (t) = x(t) - xˆ(t) , f (t) = f(t) - fˆ(t)

(13)

1

2

792

The dynamics of e (t) is given by e& (t) = x& (t) - x& (t) 1

SE ≤ SE

1

(14)

i =1

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

1

p

~ e&2 (t ) = ∑ µ i [(A i - K i C i )e 2 (t) − K i E i f (t)]

we obtain 1 V&1 = e1(t) T (H T P1 + P1H )e1(t) 2 1p ˆ T T + ∑µi [f(t) S P1e1(t)+ e1(t)T P1Sfˆ(t)- fˆ(t) SE 2 i=1

(15)

i =1

The dynamics of the fault error estimation can be writ~& & ten f (t) = f&(t) - fˆ(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 f (t) w.r.t time is

ε

With φ

1 V&1 = e1 (t) T (H T P1 + P1H + P1P1 )e1 (t) 2 1 + fˆ(t) S E − S E max fˆ(t) 2 The time derivative of V (φ (t )) is

(17)

o

(

p 0 , Bo =   , Ao = ∑ µi Aoi I  i =1

 Ai − K i Ci − K i Ei  Where A oi =   − Li Ei  − Li C i Consider the Lyapunov function candidate 1 T T V (e (t ), φ (t )) = e (t) P e (t) + φ (t) P φ (t) (18) 1 1 1 1 2 2 Where P1 and P2 are time-invariant, symmetric and positive definite matrices. Let 1 V1(e1(t)) = e1(t)TP1e1(t) , V2 (φ (t )) = φ (t)T P2φ (t) (19) 2 The time derivative of V (e (t )) is 1

2

+ f 12 λ max ( BoT Bo )

T

(

(20)

ui (t ) = BT Z −1Z ui

where Z ui = [He1 (t) + A r x(t) + Br r(t) - A i x(t) − 0.5e1 (t ) fˆ(t) S E fˆ(t) / e1 (t )T P1e1 (t ) + Sfˆ(t)

}

a symmetric T

1

exists a common positive definite matrix P1 and P2 such that H T P1 + P1H + P1P1 < 0

AoiT P2 + P2 Aoi + P2 P2 < −δI

i=1,2,…,p

(30)

From (29) and (30) 1 T T V& ≤ − e1 (t) Q e (t) − φ (t) Q φ (t) ≤ 0 (31) 1 1 2 2 If the time derivative of (18) is negative uniformly for all e1(t ),φ (t ) and for all t ≥ 0 except at

(23)

Where Z=B if B is an invertible square matrix or Z=BBT if B is not a square matrix, ⋅ denotes the l2 norm for for

)

From (28), we have 1 T T V& ( e (t ), φ (t )) ≤ − e1 (t) Q e (t) − φ (t) Q φ (t) (29) 1 1 1 2 2 e and φ converges to zero if V& < 0 . V& < 0 if there

(22)

norm

E max

2

(21)

induced

E

positive definite matrix, where δ = f1 λ max( Bo Bo ) .

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

l2

(27)

Where λmax (•) denotes the largest eigen value. Combining (25) with (27), the time derivative of V can be expressed as 1 V& (e1(t ),φ (t )) ≤ − e1(t)T Q1e1(t) − φ (t) T Q2φ (t) 2 + 0.5 fˆ(t) S − S fˆ(t) (28)

Q2 = −( AoiT P2 + P2 Aoi + P2 P2 + δ ) are

ui (t ) = Z −1Z ui

and

(26)

2

By substituting (17) into (26) and using Lemma 1 and the definition [10], one obtains p V&2 (φ (t )) = ∑ µiφ (t)T ( AoiT P2 + P2 Aoi + P2 P2 )φ (t) i =1

We design ui(t), i=1,2,…,p as follows, • When B is an invertible square matrix, the control law is given by

vectors

2

Where Q1 = -(H T P1 + P1H + P1P1) ,

- A r x(t) − Br r(t)]}

max

(25)

T T V& (φ (t )) = φ&(t) P φ (t) + φ (t) P φ& (t)

 p 1 × P1e1 (t) + e1 (t) T P1  ∑ µ i [A i x(t) + Bi u i (t) 2 i = 1

{

)

2

1

 1  p V&1 =  ∑ µ i [A i x(t) + Bi u i (t) - A r x(t) − Br r(t)] 2 i = 1 

fˆ(t) ] (24)

Using Lemma 1 to fˆ(t)T S T P1e1 (t) + e1 (t)T P1Sfˆ(t) yields

i =1

From (15) and (16), one can obtain: φ& = A φ + B f&(t)

max

Lemma1 [12]: Given constant matrices W and O appropriate dimensions for ∀ε > 0 , the following inequality holds: 1 W T O + O T W ≤ εW T W + O T O

p & ~& ~ f (t) = f&(t) - fˆ(t) = f&(t) - ∑ µi [Li Ci e 2 (t) + Li E i f (t)] (16)

o e (t ) = ~2   f (t)   

, H ∈ κ nxn is a stable matrix to be de-

signed. A block diagram of the closed-loop system is shown in Fig.1. It is assumed that Z-1 exists (Z is nonsingular). From (21) to (23) and assuming that e (t) ≠ 0 and choosing S so that S=E and SE = S T S ,

p

e&1 (t) = ∑ µ i [A i x(t) + Bi u i (t) - A r x(t) − B r r(t)]

max

matrices,

793

e (t ) = 0, φ (t ) = 0 then the controlled fuzzy system (9)

(Xi E )T + X i E + P22 P22 < −δI

1

is asymptotically stable about its zero equilibrium.

The inequalities in (34) and (35) are linear matrix inequality feasibility problems (LMIP's) in P11, P22,, Yi and Xi . By solving (34) and (35) the observer gain (Ki and Li ) can be easily determined.

x(t)

Reference Model

+

4. System model and TS fuzzy description

f(t) System

x(t) -

Observer

~y(t) -

The underlying hybrid wind-diesel system is illustrated in Fig. 2. The hybrid generation system is composed of a wind-turbine coupled with a synchronous generator, a diesel-induction generator, and an energy storage system. In the given system, the wind turbine drives the synchronous generator that operates in parallel with the storage battery system. When the windgenerator alone provides sufficient power for the load, the diesel engine is disconnected from the induction generator. The 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.

C r(t)

xˆ(t) +

fˆ(t)

C

Controller

Fig. 1: Block diagram of FTC scheme

Diesel engine

The results of this section can be summarized by the following lemma and theorem.